2020 December - CIM
Transcript of 2020 December - CIM
![Page 1: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/1.jpg)
ISSN 2183-8062 (print version)ISSN 2183-8070 (electronic version)
centro internacional de matemΓ‘tica
2020 December .42
![Page 2: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/2.jpg)
In March of 2020, a new direction board of CIM took office and the editorial board of the CIM bulletin was renewed. The editorial board is committed to continue the efforts of fulfilling the bulletinβs main goals of promoting Mathematics and especially mathematical research. The COVID-19 pandemic conditioned strongly CIMβs usual activities, which means that, contrary to previous issues, only one report of a scientific meeting supported by CIM is presented in the current issue. Nonetheless, we compensate this shortcoming with three articles regarding advances on the cutting edge of the discipline. Namely, we include an article concerning the topological properties of configuration spaces of points; an article considering stochastic processes modelling an interacting particle system, whose algebraic structure helps to analyse its macroscopic dynamics; and an article with an insightful introduction to analysis on graphs and, in particular, quantum graphs. Inserted in the cycle of historical articles, we feature an article dedicated to the work and life of AntΓ³nio Aniceto Monteiro, focusing, in particular, on his modernist essay about General Analysis, from 1939, which anticipated Bourbakiβs treaty published in the 1940s. We present an interview honouring JosΓ© Basto-GonΓ§alves, who belonged to the first scientific committee of CIM, for his role in disseminating and stimulating mathematical research in the University of Porto. We also include an interview to AndrΓ© Neves, who was the distinguished mathematician invited to deliver this yearβs Pedro Nunesβ lecture, which is an emblematic initiative of CIM, counting with the support of SPM. We recall that the bulletin continues to welcome the submission of review, feature, outreach and research articles in Mathematics and its applications.
Ana Cristina Moreira FreitasFaculdade de Economia and Centro de MatemΓ‘tica da Universidade do Portohttps://www.fep.up.pt/docentes/amoreira/
EditorialContents
02 Editorial by Ana Cristina Moreira Freitas
03 AntΓ³nio Monteiro and his Modernist Essay by JosΓ© Francisco Rodrigues
11 Where Lie Algebra meets Probability?! by Chiara Franceschini and PatrΓcia GonΓ§alves
22 An Interview with José Basto Gonçalves by Helena Reis
26 Dynamical Aspects of Pseudo-Riemannian Geometry Report by Ana Cristina Ferreira and Helena Reis
29 An Introduction to Analysis on Graphs: what are Quantum Graphs and (why) are they Interesting? by James B. Kennedy
38 An Interview with AndrΓ© Neves by Carlos Florentino
41 Configuration Spaces of Points by Ricardo Campos
2
![Page 3: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/3.jpg)
AntΓ³nio Monteiro And his Modernist essAy by JosΓ© Francisco Rodrigues*
The 1939 βensaio sobre os fundamentos da anΓ‘lise geralβ (Essay on the foundations of General Analysis), by the Portuguese mathematician AntΓ³nio Aniceto Mon-teiro (1907β1980), in spite of being awarded an import-ant prize by the Lisbon Academy of Sciences, has been unknown and ignored until its recent rediscover and its facsimile publication [AM1939]. The ensaio is a 130 pages typed monograph that introduces mathematical modernism and prepares a turning point in the mathe-matical activities in Portugal, preceding the creation of the Centro de Estudos MatemΓ‘ticos de Lisboa in 1940, the first Portuguese research centre, affiliated with the Faculdade de CiΓͺncias of the Lisbon University and independently
* Universidade de Lisboa/CiΓͺncias/CMAFcIO e Academia das CiΓͺncias de Lisboa [email protected]
supported by the Instituto para a Alta Cultura, the incipi-ent national science foundation at the time [Ro].
AntΓ³nio Monteiro, Modernist and Mathematician
On the occasion of the centenary of his birth, the Portu-guese Mathematical Society (SPM) published in 2007 a remarkable photobiography [AM_Fb2007] and a spe-cial issue of its Bulletin [AM_B2007] with the proceed-ings of an International Colloquium at the University of Lisbon. In the presentation of his Works [AM_O2008],
CIM Bulletin December 2020.42 3
![Page 4: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/4.jpg)
consisting of eight volumes of about 2800 pages, which does not include his 1939 Essay, Jean-Pierre Kahane, from the Paris Academy of Sciences, wrote
The works of AntΓ³nio A. Monteiro belong to the world history of mathematics. They cover a large variety of topics from clas-sical analysis to topology and from advanced algebra to logic in its more modern chapters. Some of them come from cours-es and synthetic presentations, but the majority of them are research papers. They are presented in different styles, occa-sionally handwritten, and also in different languages. Despite their intrinsic value, these works are a testimony of an age and of an exceptional life. They were written, in four different countries: France, Portugal, Brazil and Argentina. Monteiro was the founder of mathematical journals and various math-ematical institutions, first in Portugal, then in Latin America. He had to emigrate from Portugal because of Salazarβs regime and was also affected by the military dictatorship in Argenti-na. His life testifies the link between the struggle for science and the struggle for freedom.
Monteiro was born in MoçÒmedes (Angola) in 1907, the son of a Portuguese colonial army officer, he came to Lisbon, already orphan, to attend the Military Col-lege in 1917 before his graduation in Mathematics in 1930, at the Faculdade de CiΓͺncias, and his departure to Paris with a fellowship, where he followed courses at the FacultΓ© des Sciences and seminars at the Institut Hen-ri PoincarΓ©. In 1936, he presented his thesis, Sur lβaddi-
tivitΓ© des noyaux de Fredholm, at the University of Paris under Maurice FrΓ©chet (1878β1973) [AM_Fb2007] and [AM_O2008]. During his stay in Paris, Monteiro assumes also his mission of studying βthe organization of a Centre for Mathematical Studies which would have, among others, the objective of achieving the complete resurgence of Portuguese mathematical traditionsβ, and, in his correspondence, he even refers to the acquisition of books for the Instituto de MatemΓ‘tica [AM_Bc2007]. After his return to Lisbon, he refused to sign a com-pulsory political statement in order to be integrated at the University β Monteiro would have said: βI do not accept limitations on my intelligenceβ β and he was thus unable to pursue in Portugal the career as a mathematician he developed in his exile in 1945 in Brazil and in Argenti-na from 1950 until his jubilation and removal, also for political reasons, from the Universidad Nacional del Sur in 1975, in Bahia Blanca, where he had been Professor Emeritus since 1972 and where he died in 1980 [Re]. Between 1937 and 1943, Monteiroβs scientific and academic activity in Lisbon was carried out as a precar-ious inventor of scientific libraries in Portugal. In spite of financial difficulties, he was a major participant of the brief decade of the Portuguese Mathematical Move-ment (1936β1946), which began with the activities of
Figure 1. Cover of the photobiography [AM_Fb2007] Figure 2. Cover of [AM_Bc2007]
4
![Page 5: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/5.jpg)
the NΓΊcleo of Mathematics, Physics and Chemistry at the end of 1936, with the founding, by Monteiro, of the journal Portugaliae Mathematica in 1937 (Fig.3), with the beginning of the SeminΓ‘rio de AnΓ‘lise Geral in 1939, the creation of the Centro de Estudos de MatemΓ‘tica de Lis-boa (CEML), under his scientific direction, the Gazeta de
MatemΓ‘tica and the Portuguese Mathematical Society, all in 1940. His remarkable qualities as researcher and professor were first developed in Lisbon with the orientation of young mathematicians in the first three years of the ac-tivity of the CEML, which were marked by the influence of the modernist ideas of the 1939βs βEnsaioβ and cul-minating in the visit of Maurice FrΓ©chet in early 1942 (Fig.4), and the successive departure of the young grant-ees abroad, including the first two Portuguese disciples of Monteiro, Hugo Ribeiro for the ETH in Zurich and JosΓ© SebastiΓ£o e Silva for the University of Rome [AM_B2007]. He continued his activities at the Centro de Estudos de MatemΓ‘tica do Porto, created in 1942 and affiliated with the Faculdade de CiΓͺncias of the University of Por-to, where he had a third disciple, Alfredo Pereira Gomes who also began his academic career in France and Brazil, and where he was supported by the Junta de Investigação MatemΓ‘tica (JIM). This remarkable association was cre-ated in 1943 and was sponsored by private funds during some years. JIM aimed to bring together almost all the (few) Portuguese researchers in the country, had as its primary objective βto promote the development of math-ematical researchβ and played a very important role in funding scientific publications, particularly, the journal Portugaliae Mathematica, after the government stopped the initial financial support. This support to Portugaliae Mathematica was accomplished in connection with the CEML, and was limited to the first three volumes.
Figure 3. Frontispiece of vol. 1 of Portugaliae Mathematica.
Figure 4. M. FrΓ©chet, P. J. da Cunha and A. Monteiro at Faculdade de CiΓͺncias de Lisboa in early 1942 ( [AM_Fb2007]).
CIM Bulletin December 2020.42 5
![Page 6: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/6.jpg)
Between 1945 and 1948, Monteiro was professor of AnΓ‘lise Superior at the University of Brazil, now the Uni-versidade Federal de Rio de Janeiro, where he had a strong in-fluence on young mathematicians. These include MaurΓcio Peixoto, who was his co-author, Maria Laura Lopes, whose thesis of 1949 solved a question raised by Monteiro, and also Leopoldo Nachbin, who succeeded him in directing the series of monographs Notas de MatemΓ‘tica and was the author of its No. 4, the influential lectures notes on EspaΓ§os Vetoriais TopolΓ³gicos (1948). This series, which Monteiro had founded in 1948, was published in Rio de Janeiro until 1972 and included in its No. 5 the text on Rings of Continuous Functions by Marshal H. Stone. It was continued with volume 48 by North-Holland and reached in 2008, already with Elsevier, the number 208 of that well known collection of Mathematics Studies [Ro]. Appointed Professor at the Universidad Nacional de Cuyo, San Juan, Argentina, in 1950, Monteiro found-ed the Revista MatemΓ‘tica Cuyana with M. Cotlar and E. Zarantonello, in 1955. Although he was invited to the University of Buenos Aires, Monteiro, with some of his new Argentine disciples, moved in 1957 to the recent-ly created Universidad Nacional del Sur, in BahΓa Blanca, where he founded the Mathematical Institute, the Math-ematical Library, new series of monographs and devel-oped research on Algebraic Logic and Latices. In 1974 he was appointed an honorary member of UniΓ³n MatemΓ‘ti-ca Argentina, due to his remarkable intellectual influence. According to the Argentine mathematician Eduardo Or-tiz, Monteiro belongs βto an old tradition of Argentin-ian progressive and independent thought to which the country owes some of its most valuable achievementsβ [AM_PM1980]. Although he spent a sabbatical year in Europe, during 1969β1970, visiting several universities in France, Ro-mania, Belgium, Italy and England, he did not return to Lisbon until March 1977, with a scholarship of the Na-tional Institute of Scientific Research. There he resumed his research for about two years at the Centre for Math-ematics and Fundamental Applications (CMAF), the di-rect successor of the CEML he had directed thirty seven years ago. During this period, he supervised his fourth Portuguese disciple, M. Isabel Loureiro, and wrote the extensive work Sur les AlgΓ¨bres de Heyting SymΓ©triques, which, in 1979, was awarded the Gulbenkian Prize for Science and Technology and was published in Portugaliae Mathematica [AM_PM1980], in a volume posthumously dedicated to him. In a letter dated June 5, 1978 sent to Alfredo Pereira Gomes, his former disciple from Porto, then professor at the University of Lisbon, despite his state of health, Monteiro wrote βI am really satisfied with
the results of my scientific activity in Portugal. This is mainly due to the Centro de MatemΓ‘tica (CMAF), which provided me with free time to studyβ [AM_PM1980]. On his return to BahΓa Blanca, where he had residence with his family, Monteiro died on 29 October 1980 in the country of his second exile. In a letter to his Argen-tine friend he wrote: βThatβs life dear Ortiz. One uses and spends oneself on tasks that cannot be finished: and yet one begins with enthusiasm and dedication, because hopes and certainties are never lost. Sadnesses of Bahia Blanca! on the margins of the NapostΓ‘; between winds and storms in which the earth drowns us, I see Lisbon distant - memories of my childhood!β [AM_B2007].
The forerunner 1939βs βEnsaio sobre os fundamentos da anΓ‘lise geralβ
In the Foreword of his Ensaio Monteiro wrote:
The General Analysis was founded at the beginning of this century by Maurice FrΓ©chet, with the aim of generalising the differential and integral calculus for those functions where the independent variable β and possibly the function itself βare elements of any nature (. . .) having as goal the study of the correspondences between variables of any nature.
In fact, FrΓ©chet was a pioneer in proposing, in his 1906 thesis under the guidance of Jacques Hadamard (1865β1963), an abstract approach to mathematical analysis based on general structures: class (L), spaces with con-vergence; class (E), spaces with Γ©cart, i.e., with distance, renamed metric spaces by F. Hausdorff, in 1914, as a sub-class of topological spaces, and class (V), a generalization of spaces (E) provided with neighborhoods (voisinages). These notions were evolving and FrΓ©chet set out his re-sults in the influential book Les Espaces Abstraits [F], which strongly marked Monteiroβs early mathematical activity and, in particular, his essay. Monteiro shared the view of Norbert Wiener (1894β1964), who visited FrΓ©chet in Strasburg in 1920 and later wrote about him: βmore than anyone else who had seen what was implied in the new mathematics of curves rather than points (β¦) One of the specific things which attracted me in FrΓ©chet was that the spirit of his work [spirit of abstract formalism]β. The 1939 essay clearly reflects the βspirit of abstract formalismβ, which Monteiro absorbed from FrΓ©chet and had anticipated Bourbaki in France, but unfortunately has remained unpublished and lost in the archives of the Lisbon Academy of Sciences until recently. It consists of four chapters: Abstract Set Theory (13 p.); Abstract Al-gebra (52 p.); Abstract Topology (26 p.) and Abstract Analysis or General Analysis (38 p.) and corresponds to
6
![Page 7: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/7.jpg)
a programmatic plan, which was put into practice im-mediately by himself, with a Course in1939 and in the Seminar of General Analysis in 1940 (Fig. 5), already within the scope of the recently created Centro de Estudos MatemΓ‘ticos de Lisboa [Ro]. In the first chapter, considering the Set Theory as a chapter of General Analysis, Monteiro characterized it as the theory that βoccupies itself with the properties of the sets of points that remain invariant in relation to the group of biunivocal transformationsβ, such as Abstract Algebra,
βone of the most recent chapters of modern mathematicsβ, which deals with the properties βwhich remain invariant in relation to the group of isomorphisms (biunivocal cor-respondences which respect the operation considered)β and, in the third chapter, Topology, also a chapter of the General Analysis, as the theory βwhich studies the prop-erties of the sets of points which remain invariant in re-lation to the group of bicontinuous transformations or homeomorphisms.β Besides the influence of the French school, Monteiro, who had followed the Juliaβs seminars in Paris since 1933 until 1936 (on Groups and Algebras, Hilbert Spaces and Topology), was also well aware of the contemporary mathematical developments of the Polish and the Russian schools in Set Theory and Topology and of the German school in Algebra. Finally in the fourth and main chapter, Monteiro intro-duced βthe notion of algebraic-topological space β which we can define as a space where there is simultaneously an algebra and a topologyβ, with the aim of dealing with the
βstudy of invariant properties for a topo-isomorphism, that is, by a biunivocal correspondence that is simulta-neously a homomorphism and an isomorphismβ, espe-cially in what he called βanalytical spacesβ, that is, those for which the algebraic operation is continuous. Among these, he introduced the perfectly decomposable abeli-an topological groups, for which he proved βa theorem
of structureβ, which, being βanalogous to Banachβs and Cantor-Bernsteinβs theoremsβ (about the equivalence of two sets with the same power), establishes, in particu-lar, that if two of those groups βhave the same algebraic dimension they are topo-isomorphicβ. The new notion of algebraic dimension is a generalization of the βlinear dimension of a (vector) space of type (F) recently intro-duced by Banachβ in his classic 1932 book on ThΓ©orie des opΓ©rations linΓ©aires. Culminating a modern synthesis of some algebraic-topological structures, including topo-logical groups, normed rings and Banach spaces, Mon-teiro generalized the results of his thesis on the additivity of Fredholm kernels, obtaining necessary and sufficient conditions for the additivity of the resolvents within the rings of linear operators in Banach spaces (Fig. 6). The clarity and novelty with which the new abstract ideas are described and put into practice by Monteiro in his Ensaio is remarkable and it represents a significant progress, certainly independent and unknown to the collective of mathematicians who, under the name of N. Bourbaki, were creating the ΓlΓ©ments de MathΓ©matique which would only start publishing a year later in Paris. In his autobiography, AndrΓ© Weyl (1906β1998), one of the founders and most influential mathematicians of this collective, recorded the spirit of the time by writing [W, p.114]:
In establishing the tasks to be undertaken by Bourbaki, sig-nificant progress was made with the adoption of the notion of structure, and of the related notion of isomorphism. Retro-spectively these two concepts seem ordinary and rather short on mathematical content, unless the notions of morphism and category are added. At the time of our early work these notions cast new light upon subjects which were still shrouded in con-fusion: even the meaning of the term βisomorphismβ varied from one theory to another. That there were simple structures of group, of topological space, etc., and then also more com-plex structures, from rings to fields, had not to my knowledge
Figure 5. Announcements of the course and the seminar by A. Monteiro, already at the CEML, attached to the Faculdade de CiΓͺncias of Lisbon [AM_Fb2007].
CIM Bulletin December 2020.42 7
![Page 8: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/8.jpg)
been said by anyone before Bourbaki, and it was something that needed to be said.
That was relevant to be said, and Monteiro also knew it and wrote it very clearly not only in the Preface of his Ensaio, that he delivered the 4th February 1939 at the Academy of Sciences of Lisbon, but also throughout its four chapters, which substantial contents coincide in great portions with those of the first four issues of Bourbakiβs treaty, published in Paris in 1940 and 1942. In fact, if the initial objective of those young math-ematicians from the Γcole Normale SupΓ©rieur of Paris, who founded the Bourbaki group in 1935, was to write a new course on Differential and Integral Calculus in the form of a modern treatise on Mathematical Analysis to replace the classical Cours dβanalyse of the old French school, they evolved into an axiomatic and abstract pre-sentation of βles structures fondamentales de lβanalyseβ. The first four fascicules begin the ΓlΓ©ments de MathΓ©ma-tique: Livre I β ThΓ©orie des Ensembles (Fascicule de rΓ©sultats), 1939; Livre II β AlgΓ¨bre (Structures al-gΓ¨briques), 1942; Livre III β Topologie GΓ©nerale (Chap.I, Structures topologiques; Chap.II, Structures uniformes), 1940; Livre III β Topologie GΓ©nerale
(Chap.III, Groups topologiques; Chap.IV, Nombres rΓ©els), 1942. The 45 pages booklet on Set Theory, although dated 1939, has the printing date of February 1940, and be-gins by explaining the βmode dβemploi de ce traitΓ©β, which
βtakes the mathematics at the beginning, gives complete demonstrations and, in principle, does not suppose any particular mathematical knowledge, but only a certain habit of mathematical reasoning and a certain power of abstractionβ. In the English translation of the 1970 pro-found and enlarged revision of the Set Theory fascicule, one can read: βthe axiomatic method allows us, when we are concerned with complex mathematical objects, to sep-arate their properties and regroup them around a small number of concepts: that is to say, using a word which will receive a precise definition later, to classify them ac-cording to the structures to which they belong.β The sec-ond book, on Algebra, published in 1942, has about 160 pages and contains the first chapter of algebraic struc-tures. It is a synthesis of modern algebra which is con-sidered as a result βabove all of the work of the modern German schoolβ and recognizes the 1930 book by van der Waerden, also used by Monteiro in his 1939 Ensaio, as a source of inspiration.
Figure 6. Pages of the Essay with Monteiroβs original result [AM1939].
8
![Page 9: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/9.jpg)
Bourbakiβs third book, dedicated to General Topology, is in fact the second to be published in 1940 and con-sists of two chapters dealing with structures of another kind which βgive a mathematical sense to the intuitive notions of limit, continuity and neighbourhoodβ. The Chapter I, on topological structures, begins with open sets, to define topological space, and bases the notion of convergence on the concept of filter, obtaining the com-plete equivalence between neighborhood, open set and the topology of convergence. The Chapter II deals with uniform structures, which makes it possible to extend the structure of the metric spaces introduced by FrΓ©chet in 1906, and to generalise to the uniform spaces im-portant results, in particular of compactness and com-pleteness. Chapters III and IV of General Topology were published in 1942. Chapter III, starting with the defi-nition of a topological group, develops the theory based on filters and their convergences and on the properties of uniform structures, and concludes with some topics on topological rings and fields. Chapter IV introduces the group of the real numbers, proves the usual topological properties and basic results on series and on numerical functions, ending with an extensive and fairly complete twelve-page historical note. Those three chapters are
more innovative and have a broader scope then the cor-responding two last chapters od Monteiroβs Ensaio. However, comparing the structure of the four chapters and the respective sections of AntΓ³nio Monteiroβs essay, delivered on February 4, 1939 at the Lisbon Academy of Sciences, with the contents of these first four fascicules by Bourbaki, which are a work of another dimension and with another ambition, we are surprised by the coinci-dence of their sequencing and even by the overlapping of many of their contents. Naturally Monteiro absorbed in Paris, during his stay between 1931 and 1936, the new ideas and the most recent results of modern mathemat-ics. MonteiroΒ΄s objectives had, in a completely different scale and context, some parallelism with the ambitious programme of the Bourbaki collective, but he could not know either the plans or the contents of the ΓlΓ©ments de MathΓ©matiques. However, although Monteiro had never been a βbourbakistβ or revealed sympathies for the work of Bourbakiβs disciples, we dare to consider that his re-markable and forgotten Ensaio is, in fact, a forerunner of the great project of that collective author, which is also characterised by a remarkable modernism and structur-alism [Ro]. The influence of the contents of the Ensaio and its
CIM Bulletin December 2020.42 9
![Page 10: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/10.jpg)
author is notorious. With the intense mathematical re-search activities with a small group of students during only three years at the CEML, in Lisbon, and about one year at the CEMP, in Porto, he started together an out-line of an ephemeral Portuguese School of General To-pology, which influence extended to Rio de Janeiro. In the classic 1955 book General Topology [K], the Ameri-can mathematician J. L. Kelley, in his Foreword, not only thanks Hugo Ribeiro, but also cites in his Bibliography three articles by him and two by Monteiro, all published in Portugaliae Mathematica between 1940 and 1945, a note to the C.R. Acad. Sc. Paris by A. Pereira Gomes and the monograph in Portuguese by L. Nachbin on Topo-logical Vector Spaces, both published in 1948 [Ro].
References
[AM_PM1980] Portugaliae Mathematica, vol. 39 (1980), Edition of Sociedade Portuguesa de MatemΓ‘tica (1985), dedicated to A. A. Monteiro under the direction of Alfredo Pereira Gomes.[AM_Bc2007] Boletim da Sociedade Portuguesa de MatemΓ‘tica, NΒΊ Especial, Proceedings Centenary Conference, Faculdade de CiΓͺncias da Universidade de Lisboa, June 2007, edited by LuΓs Saraiva.[AM_Fb2007] AntΓ³nio Aniceto Monteiro, uma fotobiografia a vΓ‘rias vozes, Sociedade Portuguesa de MatemΓ‘tica, Lisboa
2007, Editors: Jorge Rezende, Luiz Monteiro and Elza Amaral.[AM_O2008] The Works of AntΓ³nio A. Monteiro, Vol.s I-VIII, Edited by Eduardo L. Ortiz and Alfredo Pereira Gomes, Fundação Calouste Gulbenkian, Lisbon and The Humboldt Press, London 2008.[AM1939] AntΓ³nio Monteiro, βEnsaio sobre os Fundamentos da AnΓ‘lise Geralβ, MemΓ³rias Academia das CiΓͺncias de Lisboa , vol. XLVII n.2 (2020), 27β169.[F] M. FrΓ©chet, Les Espaces Abstraits et leur thΓ©orie considΓ©rΓ©e comme Introduction Γ lβAnalyse GΓ©nΓ©rale, Gauthier-Villars, Paris, 1928.[K] J. L. Kelley, General Topology, Van Nostrand, New York, 1955.[Re] J. Rezende, Blogue <AntΓ³nio Aniceto Monteiro> http://antonioanicetomonteiro.blogspot.com/
[Ro] J. F. Rodrigues, O Ensaio de 1939 de AntΓ³nio Monteiro, O seu contexto e a sua importΓ’ncia, MemΓ³rias Academia das CiΓͺncias de Lisboa, vol. XLVII n.2 (2020), 9β26.[W] A. Weil, The Apprenticeship of a Mathematician, Springer, Basel, 1992.
Figure 7. The study on Continuous Functions, published by the CEM of Porto in 1944.
Figure 8. A paper on general topology by A. Monteiro, published by CEM of Lisbon in Portugaliae Mathematica, 1 (1940), 333β339, and cited in the Kelleyβs book [K].
10
![Page 11: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/11.jpg)
Where Lie ALgebrA Meets ProbAbiLity?!
by Chiara Franceschini* and PatrΓcia GonΓ§alves**
* Center for Mathematical Analysis Geometry and Dynamical Systems, Instituto Superior TΓ©cnico, Universidade de Lisboa [email protected]
** Center for Mathematical Analysis Geometry and Dynamical Systems, Instituto Superior TΓ©cnico, Universidade de Lisboa [email protected]
One of the biggest challenges in statistical physics is to understand phenomena out of equilibrium. A common setting to model non-equilibrium dynamics is to consider stochastic processes of Markovian type with an open boundary acting on the system at different values, thus creating a flux. In these notes, we consider an interacting particle system known in the literature as the symmetric simple exclusion process (SSEP) which is connected to two reservoirs. We show how the algebraic construction of such Markov jump processes helps in analyzing microscopic quantities used to derive macroscopic universal laws. In particular, we will characterize through its moments the non-equilibrium stationary measure.
1 INTRODUCTION
Microscopic dynamics of random walks interactingon a discrete space under some stochastic rules areknown as interacting particle systems (IPS) and wereintroduced in the mathematics community by Spitzerin 1970 (see [14]) but before were widely used byphysicists, see [15]. The idea of introducing such sysΒtems is that, as it often happens in mathematics andphysics, they can be used as toy models to describecomplex stochastic phenomena involving a large numΒber (typically of the order of the Avogadroβs number)of interrelated components. Regardless their simplerules at the microscopic level, IPS are often remarkΒably suitable models capable of capturing the sortof phenomena one is interested at the macroscopiclevel. Mathematically speaking, they are treated ascontinuous time Markov processes with a finite orcountable discrete state space. Typically, in the fieldof IPS one is interested in deriving the macroscopiclaws of some thermodynamical quantities by meansof a scaling limit procedure. The setting can be deΒscribed as follows. One considers a continuous space,which is called the macroscopic space. This space is
then discretized by a scaling parameter ππ and time isspeeded up by a function of ππ. On the discrete spaceone considers a microscopic dynamics consisting ofthe infinitesimal evolution of particles according tosome stochastic law. The dynamics conserves one(or more) thermodynamical quantity and its (their)space/time evolution is the object of our interest. Themathematical rigorous derivation of the macroscopiclaws for such quantity, which can be a PDE or astochastic PDE, depending on whether one is at thelevel of the Law of Large Numbers or at the level ofthe Central Limit theorem, is a central problem in thefield of IPS. This derivation gives not only validity tothe equations obtained but also some physical motivaΒtion for their study. In these notes we present, as toymodel, the most classical IPS and our aim is, first, exΒplain how to rewrite the Markovian generator of theprocess in terms of the generators of a Lie algebra, thisis a known procedure in the literature. This techniqueallows to derive a dual process for our model, whosedynamics is simpler. It can be used to give relevant inΒformation about our original model; second, explainhow to extract from our random dynamics a solutionto a PDE, describing the spaceΒtime evolution of thedensity of our model.
1
CIM Bulletin December 2020.42 11
![Page 12: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/12.jpg)
2 THE MODEL
To make our presentation simple, we consider asmacroscopic space the interval [0, 1]. Let ππ be a scalΒing parameter, we split that continuous space into inΒtervals of size 1/ππ.
To an interval of the form [π₯π₯/ππ, π₯π₯π₯ π₯ 1π₯/πππ₯ we asΒsociate the microscopic position π₯π₯ and then we havea discrete space which is the microscopic space. Nowwe define the random dynamics. Among the simplestandmost widely studied IPS there is the SSEP, see e.g.[12], whose dynamics can be described as follows. AfΒter a certain random time, a particle decides to jumpto a position of the microscopic space. In SSEP, partiΒcles jump among sites under the exclusion rule, namelyeach site can accommodate at most one particle, thereΒfore, if a particle wants to jump to an occupied site,that jump is forbidden. Interactions are to nearestΒneighbors (and this is why the process coins the namesimple) and the jump rates to left and right are idenΒtical (symmetric). Our toy model is the SSEP with anopen boundary, namely we attach two reservoirs thatcan inject or remove particles from their neighbor poΒsitions. The time between jumps is exponentially disΒtributed, which guarantees that this process is MarkoΒvian, therefore its evolution can be entirely describedvia its Markov generator. In the next subsection wedefine it rigorously.
2.1 PROBABILISTIC DESCRIPTION
Consider the microscopic space Ξ£ππ βΆ= {1, β¦ , ππ π 1π,called bulk, which corresponds to the macroscopic inΒterval [0, 1]. The construction of the SSEP evolvingon Ξ£ππ is done in the following way. To properly deΒfine the exchange dynamics, for each π₯π₯ π₯ Ξ£ππ, we callπππ₯π₯π₯π₯ the occupation variable at site π₯π₯: if πππ₯π₯π₯π₯ = 0(resp. πππ₯π₯π₯π₯ = 1) it means that site π₯π₯ is empty (resp.occupied). With this restriction, the state space of ourMarkov process is Ξ©ππ βΆ= {0, 1πΞ£ππ . We denote byππ π₯ Ξ©ππ a configuration of particles. To each bondof the form {π₯π₯, π₯π₯ π₯ 1π with π₯π₯ = 1, β¦ , ππ π π₯, weassociate a Poisson process of parameter 1, that wedenote by πππ₯π₯,π₯π₯π₯1π₯π‘π‘π₯. Now we describe the boundarydynamics. We artificially add the sites π₯π₯ = 0 andπ₯π₯ = ππ that stand for the left and right reservoirs,respectively. We associate two independent PoissonProcesses to each bond {0, 1π and {ππ π 1, πππ in thefollowing way: ππ0,1π₯π‘π‘π₯ (resp. ππππ,πππ1π₯π‘π‘π₯) with paramΒ
eter πΌπΌπππππ (resp. πΏπΏπππππ) and ππ1,0π₯π‘π‘π₯ (resp. πππππ1,πππ₯π‘π‘π₯)with parameter πΎπΎπππππ (resp. π½π½πππππ). All the Poissonprocesses described above are independent, so thatthe probability that two of them take the same valueis equal to zero. This means that only one jump ocΒcurs whenever there is a possible transition. Beforewe proceed, we note that the role of the parametersπΌπΌ, πΎπΎ, π½π½, πΏπΏ πΌ 0 is to fix the reservoirsβ density/current,while the role of ππ π₯ β is to tune the strength of thereservoirsβ according to the scale parameter ππ. Taking,for example, ππ negative the reservoirs are strong andfor ππ positive, they are weak and interactions betweenthe boundary and the bulk is weaker as the value of ππincreases.
We observe that given the initial configuration ofthe system plus the realization of all the Poisson proΒcesses, it is straightforward to obtain the whole evoluΒtion of the system. The role of the Poisson processesis to fix the random time between jumps. We showan example in figure 1 where we consider ππ = π: aninitial condition is given namely ππ0 = πΏπΏπ₯, ie the conΒfiguration with just a particle at site π₯, together withall the realizations of the Poisson processes.
In figure 2 we exhibit all the configurations that weobtained from the initial configuration ππ0 = πΏπΏπ₯ and allthe realizations of the Poisson processes given in figΒure 1.
Wewarn the reader that belowwe indexed the conΒfigurations in terms of the marks of the Poisson proΒcesses and not in time, since ourMarkov chain evolvesin continuous time.
We denote by πππ₯π₯,π₯π₯π₯1 the configuration obtainedfrom ππ by swapping the values πππ₯π₯π₯π₯ and πππ₯π₯π₯ π₯ 1π₯,that is πππ₯π₯,π₯π₯π₯1π₯π§π§π₯ = π§π§Ξ£ππ⧡{π₯π₯,π₯π₯π₯1ππ₯π§π§π₯πππ₯π§π§π₯ π₯ π§π§{π₯π₯ππ₯π§π§π₯πππ₯π₯π₯ π₯1π₯ π₯ π§π§{π₯π₯π₯1ππ₯π§π§π₯πππ₯π₯π₯π₯. On the other hand, when we seea mark of a Poisson process from the boundary, forexample, ππ0,1π₯π‘π‘π₯ (resp. ππ1,0), this means that we inΒject (resp. remove) a particle at the position π₯π₯ = 1,if this site is empty (resp. occupied), otherwise nothΒing happens. More precisely, ππ1 is the configurationobtained from ππ by flipping the occupation variable at1, that is ππ1π₯π§π§π₯ = π§π§Ξ£ππ⧡{1ππ₯π§π§π₯πππ₯π§π§π₯ π₯ π§π§{1ππ₯π§π§π₯π₯1 π πππ₯1π₯π₯π§
The exchange dynamics is described by the generΒator πΏπΏπππ₯π₯, which acts on functions ππ βΆ Ξ©ππ β β asπΏπΏπππ₯π₯πππ₯πππ₯ = βππππ₯
π₯π₯=1 πΏπΏπ₯π₯,π₯π₯π₯1πππ₯πππ₯ whereπΏπΏπ₯π₯,π₯π₯π₯1πππ₯πππ₯ = πππ₯π₯,π₯π₯π₯1π₯πππ₯ [πππ₯πππ₯π₯,π₯π₯π₯1π₯ π πππ₯πππ₯] (2)
and the rates are
πππ₯π₯,π₯π₯π₯1π₯πππ₯ = {πππ₯π₯π₯π₯π₯1 π πππ₯π₯π₯ π₯ 1π₯π₯ π₯ πππ₯π₯π₯ π₯ 1π₯π₯1 π πππ₯π₯π₯π₯π₯ππ§
212
![Page 13: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/13.jpg)
Figure 1.β An initial configuration and marks of the Poisson clocks between each bond.
Figure 3.βSchematic description of dynamics of open SSEP.
Figure 2.βConfigurations evolving according to the marks of the Poisson processes.
The left reservoir generator acts on functionsππ π πππ β β as
πΏπΏβππππππ π 1ππππ {πΌπΌπ1 πΌ πππ1ππ πΌ πΌπΌπππ1ππΌ [πππππ1π πΌ ππππππ]
and the right reservoirπΏπΏππ is defined analogously, with1 replaced by πππΌ1, πΌπΌ by πΏπΏ and πΌπΌ by π½π½. Finally, the openSSEP dynamics is described by a superposition of thetwo dynamics described above, the exchange and the
flip dynamics, so that its full generator is given by
πΏπΏSSEP π πΏπΏβ πΌ πΏπΏππππ πΌ πΏπΏππ. (3)
Observe that the left and right reservoirs at differΒent densities (respectively ππππ π πΌπΌπΌππΌπΌ πΌ πΌπΌπ and ππππ ππΏπΏπΌππ½π½ πΌ πΏπΏπ) impose a flux of particles throughout thesystem. See a picture below for an illustration of thedynamics just defined.
3CIM Bulletin December 2020.42 13
![Page 14: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/14.jpg)
2.2 ALGEBRAIC DESCRIPTION
An interesting feature of some IPS is that their generΒators can be entirely described by the generators of asuitable algebra, for our toy model this will be the Liealgebra π°π°π°π°π°π°π°. More of such constructions were introΒduced in [10] and further developed in [4]. The Liealgebra π°π°π°π°π°π°π° is a 3Βdimensional vector space of traceΒless matrices together with the bilinear map [β , β ] βΆπ°π°π°π°π°π°π° π° π°π°π°π°π°π°π° π° π°π°π°π°π°π°π° called Lie bracket, which isantiΒsymmetric, i.e. [π₯π₯, π₯π₯] = β [π₯π₯, π₯π₯] and satisfies theJacobi identity, i.e. [π₯π₯, [π₯π₯, π¦π¦]]+[π₯π₯, [π¦π¦, π₯π₯]]+[π¦π¦, [π₯π₯, π₯π₯]] =0 for all π₯π₯, π₯π₯, π¦π¦ π₯ π°π°π°π°π°π°π°. We equip π°π°π°π°π°π°π° with the adΒjoint map β βΆ π°π°π°π°π°π°π° π° π°π°π°π°π°π°π°, i.e. π₯π₯ π° π₯π₯β such that(π₯π₯β)
β = π₯π₯ and [π₯π₯β, π₯π₯β] = [π₯π₯, π₯π₯]β. Usually a basis forπ°π°π°π°π°π°π° is given by the Pauli matrices,
ππ1 = (0 11 0) , πππ° = (
0 βππππ 0 ) , ππ3 = (
1 00 β1) ,
which are hermitian and unitary. Such matrices satΒisfy the following commutator and adjoint relations[ππππ, ππππ+1] = π°ππππππ+π° and ππβ
ππ = ππππ for ππ π₯ β π°mod 3π°.We also introduce the quadratic element calledCasimir (which does not belong to π°π°π°π°π°π°π°) as πΆπΆ =πππ°
1 + πππ°π° + πππ°
3; it is central, i.e. it commutes with allthe elements of π°π°π°π°π°π°π° and it is selfΒadjoint. For ourpurpose, it will be more convenient to introduce thebasis of real operators π½π½ 0, π½π½ + and π½π½ β which we callgenerators of the Lie algebra π°π°π°π°π°π°π°. They are givenby
π½π½ β βΆ=ππ1 β πππππ°
π°, π½π½ + βΆ=
ππ1 + πππππ°π°
, π½π½ 0 βΆ=ππ3π°
,
and they satisfy the following commutation and adΒjoint relations [π½π½ 0, π½π½ Β±] = Β±π½π½ Β±, [π½π½ +, π½π½ β] = π°π½π½ 0 andπ°π½π½ 0π°β = π½π½ 0, π°π½π½ +π°β = π½π½ β. The Casimir element in thissetting is ππ = π°π°π½π½ 0π°π° + π½π½ +π½π½ β + π½π½ βπ½π½ + . Besides thematrices representation, an equivalent representationis given by the action on functions ππ βΆ {0, 1} π° β as
β§βͺβ¨βͺβ©
π°π½π½ βπππ°π°πππ₯π₯π° = π°1 β πππ₯π₯π°πππ°πππ₯π₯ + 1π°π°π½π½ +πππ°π°πππ₯π₯π° = πππ₯π₯πππ°πππ₯π₯ β 1π°π°π½π½ 0πππ°π°πππ₯π₯π° = π°πππ₯π₯ β 1/π°π°πππ°πππ₯π₯π°
where we made the convention πππ°β1π° = πππ°π°π° = 0.The operators above are also known as angular moΒmentum operators. We now show how to write theopen SSEP dynamics in this context. In particular, itis verified that the exchange generator defined in (3)can be written as the tensor product of the Casimirelement. For sites π₯π₯, π₯π₯ + 1 we get
πΏπΏπ₯π₯,π₯π₯+1 = π½π½ +π₯π₯ π½π½ β
π₯π₯+1 + π½π½ βπ₯π₯ π½π½ +
π₯π₯+1 + π°π½π½ 0π₯π₯ π½π½ 0
π₯π₯+1 β 1/π° , (4)
and then summing over Ξ£ππ we obtain πΏπΏπππ₯π₯.A similar description holds true for the generators
of the boundary reservoirs,
πΏπΏβ = 1ππππ {πΌπΌ [π½π½ β
1 + π½π½ 01 β 1
π°] + πΎπΎ [π½π½ +1 β π½π½ 0
1 β 1π°]}
and similarlyπΏπΏππ is obtained replacing the algebra genΒerators acting on site ππ β 1. Note that above the notaΒtion π½π½ ππ
π₯π₯ for ππ π₯ {0, +, β} means that the generator isacting on the occupation variable at site π₯π₯ π₯ Ξ£ππ. Whyis this algebraic description useful? In the next secΒtion we see that, whenever it is possible to describea Markov generator using an algebra representationthen a useful property, duality, can be derived.
3 DUALITY FOR MARKOV GENERATORS
The advantage of dealing with a stochastic evolutionlies in the possibility to use probabilistic techniqueswhich considerably simplify the analysis of the system.A powerful tool to deal with Markov processes is du-ality theory, see [13]. This theory allows several simΒplifications: in a nutshell, one can infer informationon a given process by using a simpler one, its dual.For our toy model, we will see how to relate the openSSEPwith a simpler systemwhere the open boundaryis turned into an absorbing boundary. Indeed, dualΒity in the context of IPS allows βreplacingβ boundaryreservoirs, modeling birth and death processes, withabsorbing reservoirs which, as time goes to infinity,will eventually absorb all the particles in the system. Itis due to this simplification that one can study properΒties such as the ππβpoint correlation function of a nonΒequilibrium system using properties of a dual systemconsisting of only ππ dual particles. The link betweenthese two processes, the original, denoted by πππ‘π‘ andwith state space Ξ©, and its dual, denoted by Μπππ‘π‘ andwith state space Ξ©Μ, is provided by a set of soΒcalledduality functions π·π· βΆ Ξ© π° Ξ©Μ π° β , i.e. a set of observΒables that are functions of both processes and whoseexpectations, with respect to the two randomness, satΒisfy the following relationship for all π‘π‘ π‘ 0
πΌπΌππ[π·π·π°πππ‘π‘, Μπππ°] = οΏ½ΜοΏ½πΌ Μππ[π·π·π°ππ, Μπππ‘π‘π°] . (5)
Above πΌπΌππ (resp. οΏ½ΜοΏ½πΌ Μππ) is the expectation with respect tothe law of the πππ‘π‘ process initialized at ππ (resp. the Μπππ‘π‘process initialized at Μππ). If the generators of the proΒcesses are explicit, denoting by β the generator of Μπππ‘π‘and by Μβ the generator of its dual, Μπππ‘π‘, then a dualityrelation with duality function π·π· translates in saying
414
![Page 15: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/15.jpg)
that
(βπ·π·π·π·π· π·π·π·π·)π·π·π·π· π( π·βπ·π·π·π·π·π· π·π·)π· π·π·π·π·π (6)
In other words, the action ofβ on the first variable ofπ·π· is equivalent to the action of π·β on the second variΒable of π·π·. This is when the algebra comes in. ProvΒing the above relation knowing just the definition ofthe original process π·π·π‘π‘ by its generator would be verycomplicated, however with the algebraic descriptionof π·π·π‘π‘ one can have a feeling of what to look for. Theidea is that we can decompose the Markov generatorusing the algebra generators as building blocks. Thissimplifies the analysis because instead of looking for aduality function, one looks for an intertwiner functionbetween two representations of the Lie algebra π°π°π°π°π·π°π·.Such intertwiner function between theπ½π½ 0π· π½π½ βπ· π½π½ + repΒresentations yields the duality function. Given thespecial features of our toy model, the duality functionwill turn out to be product of indicator functions, forwhich the direct computation is not hard. NevertheΒless, for more general dynamics which, for example,allows more than a particle per site, the duality funcΒtion has a more complicated form and the aforemenΒtioned decomposition brings advantages in the proofof the duality relationship.
3.1 DUALITY FOR OPEN SSEP
In the following result we give all the ingredients tofind a duality relation for our model: it states that,the open SSEP is dual, via a moment duality functionπ·π·, to a Markov process with the same exclusion dyΒnamics inΞ£ππ but with only absorbing reservoirs at theboundary.
THEOREM 1 (DUALITY FOR OPEN SSEP).β For theopen SSEP with generator given in 3, the duality reΒlation in (6) is verified for π·π· π· π·ππ Γ π·π·ππ β β givenby
π·π·π·π·π·π· π·π·π·π· π (ππππ)π·π·π·0
βπ₯π₯π₯Ξ£ππ
1{π·π·π₯π₯β₯ π·π·π·π₯π₯} (ππππ)π·π·π·ππ (7)
where π·π·ππ π·π β0 Γ π·ππ Γ β0 and the dual generatorπ·πΏπΏ π π·πΏπΏβ + π·πΏπΏ0 + π·πΏπΏππ acts on functions ππ π· π·π·ππ β β asπ·πΏπΏπππ₯π₯ππ π· π·π·π·π· π βππβπ°
π₯π₯ππ₯ πΏπΏπ₯π₯π·π₯π₯+π₯πππ· π·π·π·π· in the bulk, whereπΏπΏπ₯π₯π·π₯π₯+π₯is as in (2); and
π·πΏπΏβπππ· π·π·π·π· π π₯ππππ π·πΌπΌ + πΌπΌπ· π·π·π·π·π₯π· [πππ· π·π·π·π₯π· β πππ· π·π·π·π·]
and analogously for the right reservoir with πΌπΌ + πΌπΌ reΒplaced by π½π½ + π½π½.
For the interested reader, a rigorous proof of this theΒorem (for ππ π 0) can be found in [5] and it is nothard to generalize it for any value of the parameter ππ.
Note that the action in the bulk of the dual generatorhas the same dynamics of the original one, while theboundary generators only absorb particles from sitesπ₯ and ππ β π₯. The plan is to describe the dual processwith another π°π°π°π°π·π°π· and find the duality function asintertwiner function of the algebra generators. Thisis given by the following representation which act onthe same functions ππ π· {0π· π₯} β β as
β§βͺβͺβ¨βͺβͺβ©
π· π·π½π½ βπππ·π· π·π·π·π₯π₯π· π π·π·π·π₯π₯ππ π· π·π·π·π₯π₯ β π₯π·π· π·π½π½ +πππ·π· π·π·π·π₯π₯π· π π·π₯ β π·π·π·π₯π₯π·ππ π· π·π·π·π₯π₯ + π₯π· + π·π° π·π·π·π₯π₯ β π₯π·πππ· π·π·π·π₯π₯π·
β π·π·π·π₯π₯πππ· π·π·π·π₯π₯ β π₯π·π· π·π½π½ 0ππ π·π· π·π·π·π₯π₯π· π π· π·π·π·π₯π₯ β π₯/π°π·πππ· π·π·π·π₯π₯π· β π·π·π·π₯π₯πππ· π·π·π·π₯π₯ β π₯π·
where, again, πππ·βπ₯π· π πππ·π°π· π 0. One can checkthat the above generators are a representation for theLie algebra π°π°π°π°π·π°π·. Moreover, since the Casimir is irreΒducible in any representation we can describe the dualprocess in the same way as in equation (4). At thispoint one can see that
πππ·π·π·π₯π₯π· π·π·π·π₯π₯π· ππ·π·π₯π₯!
π·π·π·π₯π₯ β π·π·π·π₯π₯π·!Ξπ·π° β π·π·π·π₯π₯π·1{π·π·π₯π₯β₯ π·π·π·π₯π₯}
satisfyπ½π½ πππππ·π·π· π·π·π·π₯π₯π·π·π·π·π₯π₯π· π π·π½π½ πππππ·π·π·π₯π₯π· π·π·π· π·π·π·π₯π₯π·
for ππ π₯ {+π· βπ· 0}, i.e. πππ·π·π·π₯π₯π· π·π·π·π₯π₯π· intertwines two repreΒsentations of the π°π°π°π°π·π°π· algebra. Note that since bothπ·π·π₯π₯π· π·π·π·π₯π₯ π₯ {0π· π₯} the above function correspond to theduality function inπ·ππ. For the left reservoir generator(the right is analogous) one has to check that
(πΏπΏβπ·π· π·π·π· π·π·π·π·) π·π·π·π· π ( π·πΏπΏβπ·π· π·π·π·π· π·π·) π· π·π·π·π·π·namely that
πΌπΌπ·π₯ β π·π·π₯π·ππππ ππ π·π·π·0
ππ [1{π·π·π₯+π₯β₯ π·π·π·π₯} β 1{π·π·π₯β₯ π·π·π·π₯}]
βπΌπΌπ·π·π₯
ππππ ππ π·π·π·0ππ [1{π·π·π₯βπ₯β₯ π·π·π·π₯} β 1{π·π·π₯β₯ π·π·π·π₯}]
ππ·πΌπΌ + πΌπΌπ· π·π·π·π₯
ππππ [ππ π·π·π·0+π₯ππ 1{π·π·π₯β₯ π·π·π·π₯βπ₯} β ππ π·π·π·0
ππ 1{π·π·π₯β₯ π·π·π·π₯}] π·
which is verified since both π·π·π₯ and π·π·π·π₯ are either 0 or π₯.
4 STATIONARY PROBABILITY MEASUREAND CORRELATIONS VIA DUALITY
The open SSEP is an irreducible continuous timeMarkov process with finite state space, therefore, bya classical theoremwe know that there exists a uniquestationary measure, that we denote by πππ π π π . Whenππ π ππππ π ππππ the stationary measure of our proΒcess is an homogeneous product Bernoulli measurewith parameter ππ. Moreover, this measure is also reΒversible. Nevertheless, when the equality ππ π ππππ π ππππ
5CIM Bulletin December 2020.42 15
![Page 16: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/16.jpg)
fails, the invariant measure is no longer of productform. Heuristically speaking, the density/current atthe reservoirs has a different intensity with respect toleft/right reservoirs intensity and, therefore, there isan induction of a current flow of particles in the sysΒtem. Take, for example, πΌπΌ πΌ πΌπΌ πΌ πΌ and πΎπΎ πΌ πΎπΎ πΌ πΎ, sothat particles are injected in the system from the rightreservoir and only exit through the left one. Belowweexplain briefly how to get some information regardΒing this measure. Without loss of generality, in orderto get information about the stationarymeasure it willbe easier to consider the special case where the reserΒvoirsβ rates satisfy πΎπΎ πΌ πΎπΎπΌπΌ and πΌπΌ πΌ πΎπΎπΎπΎ. From herewe assume that this is the case. Under these condiΒtions the density of the reservoirs coincide with theirinjection rate.
4.1 APPLICATION OF DUALITY
The peculiarity of having a dual process where theboundary becomes only absorbing relies on the factthat, even if two extra sites are considered, the toΒtal mass of the dual process can only decrease duringthe time evolution. As time increases, the bulk willbecome empty and all the dual particles will eventuΒally stay either on the left or the right reservoirs. Inparticular, we now show how duality connects themoments of the initial process ππ with the absorptionprobabilities of the dual process Μππ. This is done viathe following formula
πΌπΌπππ π π π [π·π·π·πππ· Μπππ·π· πΌ
ππ
βπππΌπΌ
ππππππ πππππΎππ
ππ β Μπππ·πππ· π· (7)
where ππ πΌ βπππ₯π₯πΌπΌ Μπππ₯π₯ is the total number of dual partiΒ
cles and β Μπππ·πππ· is the probability that ππ particles areabsorbed at the left reservoir (and the remaining πππΎππgo to the right reservoir) starting from the configuraΒtion Μππ. The proof relies on the fact that, as π‘π‘ π‘ π‘, allthe dual particles will be at sites πΌ or ππ. More detailscan be found in [5] and [6]. Indeed,
πΌπΌπππ π π π [π·π·π·πππ· Μπππ·π· πΌ π·π·π·
π‘π‘π‘π‘πΌπΌππ[π·π·π·πππ‘π‘π· Μπππ·π· πΌ
π·π·π·π‘π‘π‘π‘
πΌπΌ Μππ[π·π·π·πππ· Μπππ‘π‘π·π· πΌ πΌπΌ Μππππ Μπππ‘π·πΌπ·ππ ππ Μπππ‘π·πππ·
ππ πΌ
ππ
βπππΌπΌ
ππππππ πππππΎππ
ππ β Μπππ· Μπππ‘π·πΌπ· πΌ πππ· Μπππ‘π·πππ· πΌ ππ πΎ πππ· π
Suppose we start with a dual configuration Μππ πΌ πΎπΎπ₯π₯πΎ+
πΎπΎπ₯π₯2+β¦ πΎπΎπ₯π₯ππ
, namely we choose to put a dual particle ineach site π₯π₯πΎπ· π₯π₯2π· β¦ π· π₯π₯ππ. In this case equation (7) reads
as
π·π·π·πππ· Μπππ· πΌππ
βπππΌπΎ
πππ₯π₯πππ·
which is exactly the function of our interest for the iniΒtial process πππ‘π‘. We now show how to find the 2Βpointcorrelation function via the absorption probabilitiesof two dual exclusion particles. Note that equation(7) specialized for ππ πΌ 2 and Μππ πΌ πΎπΎπ₯π₯ + πΎπΎπ¦π¦ reads
πΌπΌπππ π π π [πππ₯π₯πππ¦π¦π· πΌ ππ2
ππβπ₯π₯π·π¦π¦π·πΌπ·+ππππππππβπ₯π₯π·π¦π¦π·πΎπ·+ππ2ππβπ₯π₯π·π¦π¦π·2π· (8)
where βπ₯π₯π·π¦π¦π·πππ· for ππ πΌ πΌπ· πΎπ· 2 is the probability thatππ particles are absorbed on the left reservoir startingwith a particle in π₯π₯ and a particle in π¦π¦. Before goingto the two particlesβ problem we show how to solvethe absorption probabilities for just one particle in thesame setting.
4.2 ABSORPTION PROBABILITY FOR ONE DUALWALKER: DRUNKARDβS WALK
This is a common exercise in probability, known asthe drunkardβs walk. Recall our dual process, imagΒine that site πΌ is the drunk manβs house and site ππ isa dangerous cliff. The man is at site π₯π₯ π₯ π₯ππ and hetakes random steps to the left and to the right withthe same probability: what is his chance of escapingthe cliff? The house and the cliff are absorbing sitesin the sense that once he reaches one of them, he willstay there forever. The jump rates are described bythe dual generator οΏ½ΜοΏ½πΏSSEP. Let us call πππ₯π₯ βΆπΌ βπ₯π₯π·πΎπ· theprobability that he reaches home starting at π₯π₯. Thenobviously πππΌ πΌ πΎ and ππππ πΌ πΌ. For π₯π₯ π₯ π₯2π· β¦ π· ππ πΎ 2π₯the probability of jumping right or left is the same,πΎ/2, while if he is in πΎ (resp. πππΎπΎ), goes to πΌ (resp. ππ)with probability πΎ/π·ππππ + πΎπ· and to 2 (resp. ππ πΎ π) withthe complement probability, ππππ/π·ππππ + πΎπ·. MathematiΒcally we have to solve the following system, which isfound by conditioning on the first possible jump ofthe drunk man:
β§βͺβ¨βͺβ©
πππΎ πΌ πΎππππ+πΎ
+ ππππ
ππππ+πΎππ2
πππ₯π₯ πΌ πππ₯π₯πΎπΎ+πππ₯π₯+πΎ2
for π₯π₯ π₯ π₯2π· β¦ π· ππ πΎ 2π₯πππππΎπΎ πΌ ππππ
ππππ+πΎπππππΎ2 π
A simple computation shows that last identities canbe rewritten in such a way that πππ₯π₯ πΌ πππ·π₯π₯π· is the soluΒtion of π·β¬ππ
ππ πππ·π·π₯π₯π· πΌ πΌ, where the operator β¬ππππ acts on
616
![Page 17: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/17.jpg)
functions ππ π πππ π π πππ π β as
(β¬ππππ πππ(πππ π 1
2Ξππππ(ππππ for ππ π₯ π2π π π ππ π₯ 2ππ
(β¬ππππ πππ(1π π ππ2(ππ(2π π₯ ππ(1ππ π ππ2
ππππ (ππ(ππ π₯ ππ(1πππ
(β¬ππππ πππ(πππ₯1ππππ2(ππ(πππ₯2ππ₯ππ(πππ₯1πππ ππ2
ππππ (ππ (ππππ₯ππ(πππ₯1ππ π
From this we know that, for ππ π₯ π2π π π ππ π₯ 2π, weare looking for an harmonic function of the one diΒmensional discrete laplacian. Therefore ππππ is a polyΒnomial in ππ, i.e. ππππ π π΄π΄ππ π π΄π΄, for π΄π΄π π΄π΄ π₯ β forππ π₯ π2π π π ππ π₯ 2π.
By using the boundary conditions, we find π΄π΄ ππ₯1/(ππ π₯ 2 π 2πππππ and π΄π΄ π (ππ π₯ 1 π πππππ/(ππ π₯ 2 π 2πππππ,so that, for ππ π₯ π2π π π ππ π₯ 2π we have
ππππ π π₯ ππππ π₯ 2 π 2ππππ π ππ π₯ 1 π ππππ
ππ π₯ 2 π 2ππππ (9)
We now turn to the absorption probabilities of twoexclusion processes.
4.3 ABSORPTION PROBABILITIES FOR TWO DUALWALKERS
The idea is the same as before, we condition on thefirst possible jump and we obtain a difference equaΒtionwhich is close to a two dimensional laplacianwithsome boundary conditions. Recall that βππππ₯π₯(πππ, forππ π ππ 1π 2 is the probability that ππ particles are abΒsorbed on the left boundary starting from the configΒuration with one particle in ππ and one particle in π₯π₯. Tosimplify notation we use ππππππ₯π₯ ππ βππππ₯π₯(πππ and we neΒglect the dependence on ππ, ππ and ππ. Conditioning onthe first jump we get the following identities
β§βͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβ©
ππππππ₯π₯ π 14[πππππ₯1ππ₯π₯ π πππππ1ππ₯π₯ π ππππππ₯π₯π₯1 π ππππππ₯π₯π1]
for 1 β ππ π₯ π₯π₯ β ππ π₯ 1ππππππππ1 π 1
2[πππππ₯1ππππ1 π ππππππππ2]
for ππ β 1π ππ π₯ 2ππ1ππ₯π₯ π ππππ
1π3ππππ [ππ2ππ₯π₯ π ππ1ππ₯π₯π₯1 π ππ1ππ₯π₯π1]π 1
1π3ππππ πππππ₯π₯ for 2 < π₯π₯ < ππ π₯ 1ππππππππ₯1 π ππππ
1π3ππππ [ππππππππ₯2 π πππππ₯1ππππ₯1 π πππππ1ππππ₯1]π 1
1π3ππππ πππππππ for 2 < π₯π₯ < ππ π₯ 1ππ1ππππ₯1 π 1
2π2ππππ [πππππππ₯1 π ππ1πππ] ππππ
2π2ππππ [ππ2ππππ₯1 π ππ1ππππ₯2]ππ1π2 π 1
1πππππ ππππ2 π ππππ
1πππππ ππ1π3
πππππ₯2ππππ₯1 π 11πππππ πππππ₯2πππ π ππππ
1πππππ πππππ₯3ππππ₯1π(10)
We observe that the above identities do not dependon the choice of ππ, nevertheless, as we will see below,
the boundary conditions satisfied by ππππππ₯π₯ do dependon ππ.
As above, by introducing the operator πͺπͺππππ , that
we define below, we can write the above system ina concise form. The operator acts on functions ππ ππππ π π πππ π πππ π π πππ π β in the following way: forππ π₯ π₯π₯ π₯ π1π π π ππ π₯ 1π we have
(πͺπͺππππ πππ(πππ π₯π₯π π πππππ₯1ππ(ππ π₯ 1π π₯π₯π π ππ(ππ π 1π π₯π₯π
π πππ₯π₯π1ππ(πππ π₯π₯ π 1π π ππ(πππ π₯π₯ π₯ 1ππ₯ (πππππ₯1 π 2 π₯ πππ₯π₯π1πππ(πππ π₯π₯π
and for ππ π₯ π1π π π ππ π₯ 2π we have
(πͺπͺππππ πππ(πππ ππ π 1π π πππππ₯1ππ(ππ π₯ 1π ππ π 1π
π πππππ2ππ(ππ π₯ 1π ππ π 2ππ₯ (πππππ₯1 π πππππ2πππ(πππ ππ π 1ππ
The coefficients satisfy πππ π ππππ π 1ππππ , otherwise
ππππ π 1. We know that, conditioning on the fist jump,ππππππ₯π₯ π ππ(πππ π₯π₯π satisfies (πͺπͺππ
ππ πππ(πππ π₯π₯π π π. The aboveequation tells us that we are looking for the harmonicfunction of a two dimensional laplacian which is reΒflected if ππ π₯ π₯π₯ and deformed by a factor that dependson ππ if we are close to the boundary. A general soluΒtion for ππ π ππ 1π 2 is of the form ππππππ₯π₯ π ππππππ₯π₯(πππ ππ΄π΄ππππ π π΄π΄πππ₯π₯ π π¦π¦πππππ₯π₯ π π₯π₯ππ. The twelve unknown conΒstants are found by using the boundary conditions,the law of total probability, some geometric symmeΒtries because thewalk gives symmetric jumps and alsothe previous result regarding the drunkardβswalk. Forππ π 2, it is easy to deduce that ππππππ₯π₯(2π satisfies
β§βͺβ¨βͺβ©
πππππ₯π₯(2π ππ πππ₯π₯(1π π πππ₯1π₯π₯π₯πππππ
πππ₯2π2ππππ
πππππ(2π ππ 1πππππππ(2π ππ π
For ππ π π, it is easy to deduce that ππππππ₯π₯(ππ satisfies
β§βͺβ¨βͺβ©
πππππππ(ππ ππ ππππ(ππ π πππ₯1πππππ
πππ₯2π2ππππ
πππππππ(ππ ππ 1πππππ₯π₯(ππ ππ π
For ππ π 1, it is easy to deduce that ππππππ₯π₯(1π satisfies
β§βͺβͺβ¨βͺβͺβ©
πππππ₯π₯(1π ππ πππ₯π₯(ππ π π₯π₯π₯1πππππ
πππ₯2π2ππππ
πππππππ(1π ππ ππππ(1π π πππ₯1π₯πππππππ
πππ₯2π2ππππ
ππππππ(1π ππ 1πππππ(1π π πππππππ(1π ππ π
Where the ππππ(1π is the probability found in the previΒous section and ππππ(ππ π 1 π₯ ππππ(1π, its complement.Using the last three equations of (10) together withthe boundary conditions of each absorbed probability
7CIM Bulletin December 2020.42 17
![Page 18: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/18.jpg)
we can write for all ππ π ππ ππ π, the probability πππ₯π₯ππ₯π₯(πππin terms of the πΆπΆππβs only. Now we consider the folΒlowing identity given by symmetry arguments
πππ₯π₯ππ₯π₯(ππ π ππππππ₯π₯πππππ₯π₯(ππwhich allows immediately to identify πΆπΆ πΆπ πΆπΆπ π πΆπΆπ.Using the law of total probability:
πππ₯π₯ππ₯π₯(ππ + πππ₯π₯ππ₯π₯(ππ + πππ₯π₯ππ₯π₯(ππ π πwe get πΆπΆπ π πππΆπΆ . And, finally, using the conditionfor πππ₯π₯ππ₯π₯(ππ:
ππππ₯π₯ππ₯π₯+π(ππ π πππ₯π₯ππππ₯π₯+π(ππ + πππ₯π₯ππ₯π₯+π(πππwe find that πΆπΆ π π
(ππππ+ππππππ(ππππ+ππππππ. For sake of comΒ
pleteness we write below the explicit values of thethree absorption probabilities.
πππ₯π₯ππ₯π₯(ππ π (ππ π π₯π₯ π π + πππππ(ππ π π π π₯π₯ + πππππ(ππ π π + ππππππ(ππ π π + ππππππ
πππ₯π₯ππ₯π₯(ππ π (π₯π₯ π π + πππππ(π₯π₯ π π + πππππ(ππ π π + ππππππ(ππ π π + ππππππ
πππ₯π₯ππ₯π₯(ππ π π₯π₯(ππ + ππ + π₯π₯(ππ π ππ π ππ₯π₯π₯π₯(ππ π π + ππππππ(ππ π π + ππππππ
+ π(ππππ π ππ(π + ππ + πππππ(ππ π π + ππππππ(ππ π π + ππππππ
.
At this point one can easily find the stationary correlaΒtion by plugging the above result into equation (8).All the computations presented above are in agreeΒment with those obtained by another method, calledthe matrix ansatz product [7], where the stationarycorrelations are also found, for details we refer thereader to [8]. An intuitive representation of the dyΒnamics of two dual exclusion particles on {ππ ππ β¦ π πππis given in the picture below. This dynamics can alΒways be represented by the dynamics of a single parΒticle which is performing a symmetric random walkbut now evolving inside the two dimensional simplex.The red points are the traps where the random walkis absorbed forever. They represent the three possiΒble ways that two dual exclusion particles can be abΒsorbed in the boundary of the lattice {ππ ππ β¦ π πππ. Ifthe randomwalk reaches the vertical cathetus itmeansthe leftmost exclusion particle has been absorbed in π,while if it reaches the horizontal cathetus, the rightΒmost exclusion particle has been absorbed in ππ. Notethat one of these two events has to happen in orderthat the randomwalk hits one of the three traps. Oncethe random walk reaches one of the two cathetus itcannot leave that cathetus, since in the dynamics ofthe two exclusion one particle is already absorbed. Onthe cathethus the dynamics of the two dimensionalrandom walk is exactly the same as the one of the one
dimensional random walk with absorbing boundary,whose absorption probability is given by the drunkΒardβs walk. Note that since two exclusion particlescannot be on the same site, we removed the diagonalπ₯π₯ π π₯π₯, while the upper diagonal π₯π₯ π π₯π₯ + π representsthe sites where the two exclusion particles are neighΒbors.
We observe that these arguments can be exΒtended to higher point correlations functions likeπΌπΌπππ π π π
[πππ₯π₯πβ― πππ₯π₯ππ
] and also to higher dimensions, but forthe purposes of this article we decide to present onlythe oneΒdimensional case and the twoΒpoint correlaΒtion function.
5 THE EVOLUTION OF DENSITY
The dynamics described above in different ways, ifnot in the presence of stochastic reservoirs, would conΒserve one quantity: the number of particles. More preΒcisely, starting from a configuration πππ with ππ π ππ π πparticles, at any time π‘π‘ we would see exactly the samenumber of particles on πππ‘π‘. Adding the stochastic reserΒvoirs, this conservation law is destroyed and the goalis to see the effect at the macroscopic level of addingreservoirs to the system. We define then a randommeasure ππππ that gives weight π/ππ to each particle as
ππππ(πππ πππππ π πππ
ππππ
βπ₯π₯ππ
ππ(π₯π₯ππππ₯π₯/ππ(πππππ
which is a positive measure with total mass boundedby π. We assume that we start our process πππ‘π‘ froma measure ππππ for which the following result is true:ππππ(πππ πππππ converges, as ππ π +π, to the measureππ(πππππ π ππ(πππππππ where ππ πΆ [ππ π] π β is a meaΒsurable function. Observe that ππππ is a random meaΒsure while ππ is deterministic. The above convergenceis in the weak sense and, by the randomness of ππππ,it is also in probability with respect to ππππ, more preΒcisely, ππππ is such that, for any ππ πΏ π and any functionππ π πΆπΆ([ππ π]π it holds
limππ
ππππ(ππ πΆ |ππππ(πππ πππππ(ππ π π πππ π πππ| πΏ ππ) π π. (11)
Above πβ π β π denotes the inner product in πΏπΏπ[ππ π] andππππ(πππ πππππ(ππ π denotes the integral of ππ with respect tothe measure ππππ(πππ πππππ. The goal is then to show thatthe same result holds true at any later time π‘π‘, but thelimit measure is given by πππ‘π‘(πππππ π ππ(π‘π‘π πππππππ, wherethe density ππ(π‘π‘π πππ is the solution of a PDE. This resultis known in the literature as hydrodynamic limit andthe PDE is the hydrodynamic equation. In the case of
818
![Page 19: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/19.jpg)
the open SEEP we have the following result.
THEOREM 2 (HYDRODYNAMICS FOR SSEP).βStarting from ππππ as described above i.e. satisfying(11) for a certain measurable function πππππ ππ π β;the trajectory of random measures ππππ
π‘π‘ (πππ‘π‘ππ2 π πππππ conΒverges, as ππ π ππ, to the trajectory of determinΒistic measures given by πππ‘π‘(πππππ π ππ(π‘π‘π πππππππ, whereππ(π‘π‘π πππ is the unique weak solution of the heat equaΒtion πππ‘π‘ππ(π‘π‘π πππ π ππ2
ππππ(π‘π‘π πππ starting from ππ and with:
β’ Dirichlet boundary conditions ππ(π‘π‘π ππ π ππππ andππ(π‘π‘π ππ π ππππ, for any π‘π‘ π‘ π, when ππ π π;
β’ Robin boundary conditions ππππππ(π‘π‘π ππ π (ππ ππΎπΎπ(ππ(π‘π‘π πππΎπππππ and ππππππ(π‘π‘π ππ π (ππππππ(πππππΎππ(π‘π‘π πππ,for any π‘π‘ π‘ π, when ππ π π;
β’ Neumann boundary conditions ππππππ(π‘π‘π ππ πππππππ(π‘π‘π ππ π π, for any π‘π‘ π‘ π, when ππ π‘ π.
The proof of the previous theorem, by using the enΒtropy method developed in [11], can be seen in [1] forthe regime ππ π π and in [2] for the regime ππ π π. Weobserve that above the time scale has been reΒscaledto π‘π‘ππ2, which is the time scale for which the evolutionof the density is nonΒ trivial, known as diffusive timescale. What if one takes shorter time scales of the formπππ π with π π π 2? Then we do not see any space/timeevolution of ππ(π‘π‘π ππππ As a consequence of the previousresult we see that on a strong action regime of thereservoir dynamics, the density profile is fixed at theboundary; while on the weak action regime, the spacederivative (current) of the profile becomes fixed.
6 HYDROSTATICS AND CORRELATIONFUNCTIONS
The reader now might ask about the stationary meaΒsure. Can we obtain the previous result starting fromthe measure πππ π π π ? This result is know in the literatureas hydrostatic limit and to recover it from last theoremone just has to derive (11) for a certain function ππ. Thecandidate is exactly the stationary solution of the corΒresponding PDE, which in the cases above is of theform Μππ(πππ π ππππ π ππ, where ππ and ππ are fixed by theboundary conditions. To prove the result we need twothings:
1. Define for π₯π₯ π₯ π₯ππ the discrete profile πππππ‘π‘ (π₯π₯π π
πΌπΌππππππππ‘π‘ππ2(π₯π₯ππ and extend it to the boundary by setΒ
ting πππππ‘π‘ (ππ π ππππ, ππππ
π‘π‘ (πππ π ππππ. Taking ππππ π πππ π π π π
we need to know that the stationary discrete proΒfile ππππ(π₯π₯π is close to Μππ( π₯π₯
πππ. One way to do it is
fromKolmogorovβs equation, inwhich one findsthat it solves the equation
πππ‘π‘πππππ‘π‘ (π₯π₯π π (π₯ππ
ππ πππππ‘π‘ π(π₯π₯π π π₯π₯ π₯ π₯ππ π π‘π‘ π π
where the operator π₯ππππ was defined in previΒ
ously.
Observe that the above equation is closed interms of ππππ
π‘π‘ (β π, this is a consequence of the factthat the generator of the dynamics does not inΒcrease the degree of functions. A simple compuΒtation allows to derive the stationary solution ofthe previous equation and to show that it is closeto Μππ(β π. Alternatively, we could use the results weobtained by duality which give
πΌπΌπππ π π π πππ(π₯π₯ππ π ππππβπ₯π₯(ππ π ππππβπ₯π₯(πππ (12)
From (9) we conclude that
πππππ π π π (π₯π₯π π ππ πΎ ππ
2ππππ π ππ πΎ 2π₯π₯π ππ πΎ ππ
2ππππ π ππ πΎ 2(πππππΎππππππ
(13)from where we can easily check that
limπππππ
maxπ₯π₯π₯π₯ππ
|πππππ π π π (π₯π₯π πΎ Μππ(π₯π₯
πππ| π ππ
2. We need to study the behavior of the twoΒpointcorrelation function defined generally by
πππππ‘π‘ (π₯π₯π π₯π₯π π πΌπΌππππ
π Μπππ‘π‘ππ2(π₯π₯π Μπππ‘π‘ππ2(π₯π₯πππwhere Μπππ‘π‘ππ2(π₯π₯π π πππ‘π‘ππ2(π₯π₯π πΎ ππππ
π‘π‘ (π₯π₯π and show that,for ππππ π πππ π π π , it vanishes as ππ π ππ. As for thediscrete profile, we can also apply Kolmogorovβsequation and derive a discrete equation for theevolution of this function and then obtain its staΒtionary solution. Alternatively, we could use theresults we obtained by duality from where wecan get the explicit expression for the stationarycorrelations. A simple, but long, computationshows that
πππππ π π π (π₯π₯π π₯π₯π π πΌπΌππ
πππ π π π π Μππ(π₯π₯π Μππ(π₯π₯ππ π
πΎ (ππ πΎ πππ2(π₯π₯ π ππππ πΎ ππ(ππ πΎ π₯π₯ π ππππ πΎ ππ(2ππππ π ππ πΎ 2π2(2ππππ π ππ πΎ ππ
from where we conclude that
maxπ₯π₯ππ₯π₯
|πππππ π π π (π₯π₯π π₯π₯π| ππππππ ππ
Even if for hydrostatics it is enough to know the orderof decay of ππππ
π π π π (π₯π₯π π₯π₯π, thanks to the approach shownabove we were able to write the actual form of thetwo point correlation function. We note that from theprevious identity we can obtain the following relationΒ
9CIM Bulletin December 2020.42 19
![Page 20: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/20.jpg)
ship
πππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ π₯ (π½π½ π₯ π½π½π₯2
2ππππ + ππ π₯ ππππ₯π₯(0π₯πππ₯π₯(1π₯
where πππ₯π₯(1π₯ is given in (9). We also note that forππ π₯ 0 the above identity becomes
πππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ π₯(π½π½ π₯ π½π½π₯2
ππ π₯ 1πΊπΊDir
(π₯π₯ππ
π₯ π₯π₯ππ)
whereπΊπΊDir(π’π’π₯ π’π’π₯ π₯ π’π’(1π₯π’π’π₯ is theGreen function of the2Βdimensional laplacian on {(π’π’π₯ π’π’π₯ π’ 0 π’ π’π’ π’ π’π’ π’ 1π’π₯reflected on the line π’π’ π₯ π’π’ and with homogeneousDirichlet boundary conditions, that is πΊπΊDir(π’π’π₯ π’π’π₯ is thesolution of
Ξπ π πΊπΊDir(π’π’π₯ π’π’π₯ π₯ π₯π’π’π’π’π₯π’π’
where for π’π’ π’ π’π’,Ξπ π πΊπΊDir(π’π’π₯ π’π’π₯ π₯ π’π’2
π’π’πΊπΊDir(π’π’π₯ π’π’π₯ + π’π’2π’π’πΊπΊDir(π’π’π₯ π’π’π₯
and for π’π’ π₯ π’π’,Ξπ π πΊπΊDir(π’π’π₯ π’π’π₯ π₯ π’π’π’π’πΊπΊDir(π’π’π₯ π’π’π₯ π₯ π’π’π’π’πΊπΊDir(π’π’π₯ π’π’π₯
and πΊπΊDir(0π₯ π’π’π₯ π₯ πΊπΊDir(π’π’π₯ 1π₯ π₯ 0. We get the scalingform
limπππ+π
πππππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ π₯(π½π½ π₯ π½π½π₯2πΊπΊDir(π’π’π₯ π’π’π₯
for the continuous correspondents π₯π₯ππ
π π’π’ and π₯π₯ππ
π π’π’.We now see what happens for the cases when ππ π’ 0.For 0 < ππ < 1, a simple computation shows that thelimit above also holds. For ππ π₯ 1 we get
limπππ+π
πππππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ π₯(π½π½ π₯ π½π½π₯2
9πΊπΊRob(π’π’π₯ π’π’π₯
where πΊπΊRob(π’π’π₯ π’π’π₯ π₯ 1π(π’π’ + 1π₯(2 π₯ π’π’π₯ and corresponds
to the Green function of the 2Βdimensional laplaciandefined above, but with homogeneous Robin boundΒary conditions given by π’π’π’π’πΊπΊRob(0π₯ π’π’π₯ π₯ πΊπΊRob(0π₯ π’π’π₯ andπ’π’π’π’πΊπΊRob(π’π’π₯ 1π₯ π₯ π₯πΊπΊRob(π’π’π₯ 1π₯. Finally, for ππ π 1, if weuse the same scaling as above, we see that
limπππ+π
πππππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ 0π₯
Nevertheless, a simple computation shows that
limπππ+π
πππππππππ π π π (π₯π₯π₯ π₯π₯π₯ π₯ π₯(π½π½ π₯ π½π½π₯2
8(π’π’ + 1π₯(1 π₯ π’π’π₯
for the continuous correspondents π₯π₯π₯ππππ π π’π’ andπ₯π₯π₯ππππ π π’π’; and this is the correct order to see a nonΒtrivial limit in the case of very slow boundary. Forhigher point correlation functions, we can use exactlythe same argument as above in order to obtain theexact rates of convergence of the corresponding staΒtionary correlations. Moreover, we conjecture that wecan write the stationary ππΒth point correlation funcΒtionππππ
π π π π (π₯π₯1π₯ β¦ π₯ π₯π₯πππ₯ as a product between a scaling facΒ
tor and the absorption probabilities of ππ independentoneΒdimensional random walks. We also believe thatthis argument could be extended to other models forwhich duality is known but all this is left for a futurework.
Recently, it has been developed a method in [9] toderive the hydrodynamic and the hydrostatic limits inpresence of duality for a similar model called the symΒmetric inclusion process, wheremany particles can ocΒcupy the same site and show a preference of layingtogether. The macroscopic behavior for this processis the same as the one described above, but the proofnow boils down to the sole use of duality.
We conclude by saying that there are many othermodels for which one has to explore the notion of duΒality, specially for asymmetric models where the equaΒtions for correlation functions are no longer closed.There is a long and standing work to develop aroundthese problems and here we just collected some niceand simple results for a toy model where Lie algebraand, consequently, duality allows getting a lot of releΒvant information about our model.
ACKNOWLEDGEMENTS
The authors thank FCT/Portugal for support throughthe project UID/MAT/04459/2013. This project hasreceived funding from the European Research CounΒcil (ERC) under the European Unionβs Horizon 2020research and innovative programme (grant agreeΒment no715734).
REFERENCES
[1] Baldasso, R., Menezes, O., Neumann, A., Souza,R. R. Exclusion process with slow boundΒary. Journal of Statistical Physics, 167(5):1112β1142, 2017.
[2] Bernardin, C., GonΓ§alves, P., JimΓ©nezΒOviedo,B. Slow to fast infinitely extended reservoirsfor the symmetric exclusion process with longjumps. Markov Processes and Related Fields, (25):217β274, 2019.
[3] Bernardin, C., GonΓ§alves, P., JimΓ©nezΒOviedo,B. A microscopic model for a one parameΒter class of fractional laplacians with dirichletboundary conditions. to appear in ARMA, 2020+,
1020
![Page 21: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/21.jpg)
CIMβs new direction board
Treasurer
JoΓ£o Gouveia Univ. Coimbra
Vice-President
Ana Cristina Moreira FreitasUniv. Porto
Secretary
Ana LuΓsa CustΓ³dio New Univ. Lisboa
Vice-President
AntΓ³nio Caetano Univ. Aveiro
President
Isabel Narra FigueiredoUniv. Coimbra
https://link.springer.com/article/10.1007 /s00205-020-01549-9.
[4] Carinci, G., Franceschini, C., GiardinΓ , C.,Groenevelt, W., Redig, F. Orthogonal dualΒities of Markov processes and unitary symmeΒtries. SIGMA. Symmetry, Integrability and Geom-etry: Methods and Applications, 15(5):, 2019.
[5] Carinci, G., GiardinΓ , C., Giberti, C., Redig, F.Duality for stochastic models of transport. Jour-nal of Statistical Physics, 152(4):657β697, 2013.
[6] Carinci, G., GiardinΓ , C., Redig, F. Consistentparticle systems and duality. arXiv:1907.10583,2019.
[7] Derrida, B., Evans, M. R., Hakim, V. andPasquier, V. Exact solution of a 1Βd asymmetΒric exclusion model using a matrix formulation.Journal of Physics: A Mathematical and GeneralPhysics, 1993.
[8] De Paula, R. Porous Medium Model in conΒtact with Reservoirs. PUCΒRio Master Thesisin Mathematics, 2017.
[9] Franceschini, C., GonΓ§alves, P., Sau, F.Symmetric inclusion process with slowboundary: hydrodynamics and hydrostatΒics. arXiv:2007.11998, 2020.
[10] GiardinΓ , C., Kurchan, J., Redig, F., Vafayi, K.Duality and hidden symmetries in interactingparticle systems. Journal of Statistical Physics, 135(1):25β55, 2009.
[11] Guo, M. Z., Papanicolaou, G. C., Varadhan, S.R. S. Nonlinear diffusion limit for a systemwith nearest neighbor interactions. Comm.Math.Phys. 118 (1) 31β59, 1988.
[12] C. Kipnis and C. Landim. Scaling limits of inter-acting particle systems. SpringerΒVerlag, 1999.
[13] T. Liggett. Interacting Particle Systems. SpringerΒVerlag, 1985.
[14] F. Spitzer. Interaction of markov processes. Ad-vances in Mathematics, 5:246β290, 1970.
[15] H. Spohn. Large scale Dynamics of Interacting Par-ticles. SpringerΒVerlag, 1991.
11CIM Bulletin December 2020.42 21
![Page 22: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/22.jpg)
an interview with
José Basto Gonçalves
* Centro de MatemΓ‘tica e Faculdade de Economia da Universidade do Porto [[email protected]]
by Helena Reis*
José Agostinho Basto Gonçalves (born 28 January 1952) graduated in Mathematics at the University of Porto in 1975 and in 1981 he received his PhD degree in Mathematics from the University of Warwick. He returned to Porto and played a massive role in the creation of a scientific culture in the Math. Department, helping to instill a research-oriented mentality in several generations of students.His main research work lies in the scope of control theory and of the geometric theory of differential equations. He became Full Professor of the University of Porto in 1991 and he retired in 2008. Over the course of his career, he has supervised two PhD theses and has mentored a number of Master and undergraduate students. He was member of the first Scientific Committee of CIM (1996-2000) and member of the Statutory Audit Committee from 2000 to 2004. He has also been president of the northern regional direction of SPM.
Photo by Isabel Labouriau
22
![Page 23: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/23.jpg)
Can you tell us in which moment you realized that you wanted to be a Mathematician and also let us know how this happened?I did not think about it, but had always assumed, even as a child, that my work would be computations or something similar, and without a fixed timetable. I was very fortunate to get all that! I initially entered an Engineering course, thinking that the Maths course was only for high school teachers, and Engineering was the course with more Maths in it. But after two years I changed to Mathematics.
You have graduated from Warwick. What were your reasons to choose this university? Also, was Warwick your first choice or have you considered other universities as well?I attended a course at ICTP (Trieste) organized by prof. Markus from Minnesota and Warwick and prof. Olec from the Academy of Sciences (I think) in Czechoslovakia, and met prof. Pritchard from Warwick. Also, my friends Luisa MagalhΓ£es and Eugenia SΓ‘ were doing their Ph.D. at Warwick so . . .
You were one of the first people in Porto to have gone to Warwick for graduate school and since then, many others have followed this path some of them under your recommendation. Do you somehow feel to have a scouting job for Warwick?Perhaps I was enthusiastic about my time at Warwick, it had been extraordinary for me; also it was easier to recommend people, and I knew well the conditions there. I always thought that it would be better if people went to different universities, but the knowledge of previous students is very important for the decision to leave Porto and study abroad.
Did you return to Porto immediately after defending your Phd thesis in Warwick? Can you describe the situation of the research in Mathematics in Portugal β and more precisely in the north region β at that time?I returned to Porto in 1981 after 4 years at Warwick. As far as I remember there were no scientific papers in Mathematics published in Porto before 74, but when I came back the level of teaching was very different from my experience before leaving (I was in engineering for two years and then did Applied Mathematics). Before I left, the Applied Mathematics group was at best very old fashioned. At the end of the 70βs, the presence of Ricardo Lima was very influential at the group, and later in
the University, through new people that studied for a Ph.D. thanks to him. He taught interesting courses, talked about research and helped former students to obtain contacts abroad and financial support. In 81 the ambient in Applied Mathematics was very good: Manuel RogΓ©rio Silva, Teresa e Pedro Lago had already returned, there was no great interest in proper Mathematics but things were moving, research was being done and there was enthusiasm. In Pure Mathematics there was no published research in the beginning of the 80βs, but the teaching was up to date. I think that this was already a fact even before 1974. Certainly, anyone studying now (or since the end of the 80βs) in Porto for a degree should be able to finish a Ph.D. anywhere. I was very fortunate in many aspects: I was in engineering to begin with, I learned a lot more physics than is now common (I did not like that at the time but it was useful!), I studied topics In Mathematics courses that were old fashioned then, but were very considered later, and my final year in the Mathematics degree was 74/75 when the list of courses had a great change.
By the time you returned to Portugal, other young Portuguese mathematicians were also returning home from graduate school. How did you get organized to foster the creation of culture of research with high standards around this time? For example, you used to have local collaborators in research or have you actually put direct effort into interacting with other colleagues from Portuguese universities?There was no organization but certainly a mutual interest: discovering the others and discussing the new results and topics. Our research budget in 1982 was something like 150 euros, the Fundação Calouste Gulbenkian and the director Alberto Amaral were an important help, later JNICT and then INIC had a complete change: we had a project with the money to invite very good mathematicians to give two-week courses, had a very generous budget to get equipment and to face the daily expenses and a number of young beginner mathematicians.
At the University of Porto, mathematicians are spread around several faculties, how does this affect research and teaching, and how have you dealt with this?I have studied Applied Mathematics and always worked in the department of Applied Math; it was much improved with time due to the efforts of its members. But the University
CIM Bulletin December 2020.42 23
![Page 24: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/24.jpg)
had and has more than a dozen departments (or similar structures) responsible for Mathematics classes, and the Science Faculty had a Pure Mathematics department as well. I never liked this situation, even when personally very convenient. We have tried to encourage collaboration within the university and endeavoured to surpass the inconveniences this causes, first in CMAUP, then in CMUP, after the two centres merged, by having all mathematicians together at the same research centre; the two departments in the Science Faculty are now just one department, but the problem persists at the University and it should not.
How easy was it in the β70s and β80s to get a grant to study/travel abroad? When and how did this change?In 1975 the number of grants was very small, INIC and Gulbenkian and NATO altogether had fewer than 100 for all sciences and humanities (or at least this was commonly said) at Ph.D. level. The situation with Mariano Gago as head of
JNICT was completely changed: now FCT has more than 1500 grants for Ph.D.
How did you obtain funding for research through your career? How important was it for your research?I studied for a Ph.D. in England with a Gulbenkian grant. The first budget I had from INIC after returning was less than 200 euros a year, that when everything was missing in the department (books, journals etc.); at that time Gulbenkian was a great help, with money for books, journals, for attending conferences. The community of active Portuguese mathematicians was very small, travelling was expensive and to stay abroad was even more expensive, all communications were done by letter through standard post service, there was still no e-mail or internet . . . Young people could get a job and work with us (now is a lot harder) but then everything was missing: there were very few books or journals, first we shared a personal computer bought with University research money, then two
Photo by Isabel Labouriau
24
![Page 25: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/25.jpg)
computers . . . We were a small group, six or seven, but we also shared with other people in the University. Again this was completely changed still in the 80βs, first with INIC and afterwards with Luis MagalhΓ£es at FCT. We were able to invite scientists for giving courses lasting one or two weeks, everybody could go to a conference per year, our library was quite good, we had computers and printers, and there was very little bureaucracy. A paradise! Funding was a constant worry in the beginning of the 80βs, but that did not affect research much: with money the effort was less, there were more people involved, for me it just was easier but for younger people good funding was fundamental.
You were member of the first Scientific Committee of CIM (1996-2000) whose goal was to develop and promote the mathematical research in Portugal. What was the role of CIM in those years and how were the measures implemented?I had not a clear idea of what should be the role of CIM, but I thought it was a good idea and could be developed without a lot of money (that of course did not exist).
What sort of progress have you detected/felt? Were they clear right from the beginning or they gradually become clear in the years to come?I am not a big believer in the power of an institution. Having a permanent teaching/research staff is of course not indispensable but not having it does not help.
Going back to the question about sending students abroad, I am aware that you consider important β if not absolutely indispensable β for young mathematicians to acquire international experience (by the way, as your former student, I remember to have my βwrists slappedβ for staying in Portugal for graduate school).
The mathematical community was very small, the number of research papers was almost zero in Porto, it was important for the students to have a different view and contact with much better research environment.
Do you think nonetheless that acquiring this international experience used to be more important years ago and/or consider that significant changes have occurred and that nowadays this type of experience is somehow less relevant?Now the number of people involved, in Portugal, is completely changed, the international relations exist and
work, it is not as necessary to go abroad to change. However, doing everything, first degree to PhD in the same place is still not a good idea.
Besides the scientific connection with England, you also have many contacts in Brazil, where several Portuguese mathematicians, especially from Porto, have obtained their Phd degrees . . . Would you comment on the role the collaboration with Brazil played in your career as well as in the evolution of the research at CMUP?I learned a lot in USP β S. Carlos, and (very) slowly moved from Control to more standard mathematics; this was only possible thanks to people from all the world I met there at the SΓ£o Carlos Workshops on Singularities (every two years) and at the university. And I have made very good friends . . .
Today CMUP is a top center for Mathematics recognized both at international and national levels. What is the feeling that such an evolution brings to you and the colleagues from your generation given that you have been the initial promoters of the culture of research? Are you especially proud of the work accomplished?I am very happy seeing that the new normal was almost unthinkable when I began. Like a coach, I expect the new ones to do better than I did!
Our research centre CMAUP went from good to excellent, many students finished their Ph.D. here or abroad, CMUP is now excellent, things are much better than they were.
Do you have hobbies or other regular interests outside the academic community?I should have thought about that long ago . . .
I would like to close the interview with a comment rather than with a question. I would like to make clear that Professor JosΓ© Basto played an important role in my Mathematical education. Namely, you were the instructor of 5 courses I have enrolled in over my undergraduate and Master courses. In addition, you have supervised my Master dissertation as well as my Phd thesis. I am very much indebted with you for everything I have learned and I also thank you for having persuaded me (finally after my thesis defenses) to go abroad for a post-doc in France . . . it certainly was very important in my life.
CIM Bulletin December 2020.42 25
![Page 26: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/26.jpg)
report
by Ana Cristina Ferreira* and Helena Reis**
Dynamical Aspects ofPseudo-Riemannian Geometry
* Centro de MatemΓ‘tica e Escola de CiΓͺncias da Universidade do Minho [[email protected]]** Centro de MatemΓ‘tica e Faculdade de Economia da Universidade do Porto [[email protected]]
The conference Dynamical Aspects of Pseudo-Riemannian Geometry was held at the School of Sciences of the University of Minho from 2 through 6 March 2020. The event has received financial support from the following institutions: Centro de MatemΓ‘tica da Universidade do Minho (CMAT), Centro de MatemΓ‘tica da Universidade do Porto (CMUP), Centro Internacional de MatemΓ‘tica (CIM), Fundação Luso-Americana para o Desenvolvimento (FLAD), Fundação para a CiΓͺncia e Tecnologia (FCT), Institut de MathΓ©matiques de Toulouse (IMT)and UniversitΓ© of Luxembourg (UNI.LU).
26
![Page 27: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/27.jpg)
Mini-Courses
François Béguin (Univ. Paris 13)Geometry and dynamics of spatially homogeneous spacetimes
JoΓ£o Pimentel Nunes (IST)Quantization and Kahler Geometry
Abdelghani Zeghib (CNRS - ENS Lyon)Configuration spaces: Geometry, Topology, Dynamics, Physics and Technology
The event consisted of an international conference revolving around dynamical systems naturally arising in important problems belonging to the field of pseudo-Riemannian geometry and, in particular, of Lorentzian geometry. In this way, the conference has attracted interest from experts in dynamical systems and geometry as well as from certain physicists. The conference has brought together more than 40 experts in the mentioned areas coming from various countries and including several field leaders for a
program consisting of 3 advanced mini-courses, 12 invited talks and a poster session. The program also included some exercise sessions, related to the material covered in the mini-courses, and an open problem session. For further information check the link https://cmup.fc.up.pt/Pseudo-Riemannian-Geometry/ Thanks to the generous support provided by our sponsors, several graduate students and post-docs were able to attend the conference and the mini-courses.
CIM Bulletin December 2020.42 27
![Page 28: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/28.jpg)
Scientific Committee
Ana Cristina FerreiraCentro de MatemΓ‘tica & Escola de CiΓͺncias da Universidade do Minho
Julio RebeloInstitut de MathΓ©matiques de Toulouse
Helena ReisCentro de MatemΓ‘tica & Faculdade de Economia da Universidade do Porto
Jean-Marc SchlenkerUniversitΓ© du Luxembourg
Abdelghani ZeghibCentre National de Recherche Scientifique (CNRS) & ENS-Lyon
Organizers
Ana Cristina FerreiraCentro de MatemΓ‘tica & Escola de CiΓͺncias da Universidade do Minho
Helena ReisCentro de MatemΓ‘tica & Faculdade de Economia da Universidade do Porto
Invited Speakers
Ilka Agricola (Univ. Marburg)Einstein manifolds with skew torsion
Thierry Barbot (Univ. Avignon)Conformally flat Lorentzian spacetimes and Anosov representations
Alexey Bolsinov (Univ. Loughborough)Integrable dynamical systems on Lie algebras and their applications in pseudo-Riemannian geometry
Marie-Amelie Lawn (Imperial College London)Translating solitons in Lorentzian manifolds
Daniel Monclair (Univ. Paris-Sud)Gromov-Thuston spacetimes
Vicent Pecastaing (Univ. Luxembourg)Pseudo-Riemannian conformal dynamics of higher-rank lattices
Miguel Sanchez (Univ. Granada)Lorentzian vs Riemannian completeness and Ehlers-Kundt conjecture
Andrea Seppi (Univ. Grenoble)Examples of four-dimensional geometric transition
Rym Smai (Univ. Avignon)Anosov representations and conformally flat spacetimes
Peter Smillie (Caltech PMA)Hyperbolic planes in Minkowski 3-space
Andrea Tamburelli (Univ. Rice)Polynomial maximal surfaces in pseudo-hyperbolic spaces
JΓ©rΓ©my Toulisse (Univ. Nice)Maximal surfaces in the pseudo-hyperbolic space
28
![Page 29: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/29.jpg)
An introduction to AnALysis on grAPhs: WhAt Are QuAntuM grAPhs And (Why) Are they interesting?
by James B. Kennedy*
* Departamento de MatemΓ‘tica and Grupo de FΓsica MatemΓ‘tica, Faculdade de CiΓͺncias, Universidade de Lisboa [email protected]
Part of the work of the author relevant to this topic was supported by the Fundação para a CiΓͺncia e a Tecnologia, Portugal, via the program Investigador FCT, reference IF/01461/2015, and project PTDC/MAT-CAL/4334/2014, as well as by the Center for Inter-disciplinary Research (ZiF) in Bielefeld, Germany, in the framework of the cooperation group on Discrete and continuous models in the theory of networks.
π£π£1
π£π£2
π£π£3
π£π£4 π£π£5
ππ1
ππ2
ππ3 ππ4
ππππ£π£1)
ππππ£π£2)
ππππ£π£3)
ππππ£π£4) ππππ£π£5)
ππ1
ππ2
ππ3 ππ4 π₯π₯π¦π¦
[0, β1]
[0, β4][0, β3]
[0, β2]
Figure 1: A simple graph on 5 vertices and 4 edges (left); to define a function ππ on a discrete realisation of the graphwe specify the values at the vertices β this also gives rise naturally to difference operators (centre); one may insteadidentify each edge ππππ with an interval [0, βππ] β β (or a halfΒline [0, +β)), and βglue togetherβ the intervals at theirendpoints in the right way, to form a metric graph (right). Here a path between two points π₯π₯ and π¦π¦ is marked inred.
X
X
Figure 2: The Cheeger cut of a graph with 20 vertices.
1 INTRODUCTION: TWO COMPLEMENTARYTYPES OF GRAPHS
Probably everyone has at least an intuitive notion ofwhat a graph is: a collection of vertices, or nodes,joined by edges. Most mathematicians, perhaps evensome nonΒmathematicians, probably have some ideaof the role that graphs play in modelling phenomenaas diverse as fine structures such as crystals and carΒbon nanostructures, social networks, the PageRankalgorithm, data processing and machine learning, β¦,but may not be so familiar with the details.
Generally speaking, at amathematical level, we areinterested in some process taking place on the graph,such as described by a difference or differential equaΒtion. The mathematics behind such equations comΒbines ideas from graph theory (obviously), linear alΒgebra, functional analysis and the theory of differΒential equations, operator theory, and mathematicalphysics; yet many of the details seem to be largely unΒknown to the wider mathematical community. As atest: do you know what quantum graphs are?
Our goal here is to give somewhat uneven introΒduction to analysis on graphs: we first describe, inhopefully accessible terms, what this is: how to deΒfine functions and difference and differential operaΒ
tors on graphs, and study them β and in particularwhat are quantum graphs. Our starting point is thatthere are (at least) two natural, somewhat parallel, noΒtions of graphs: discrete and metric graphs; the forΒmer give rise to difference operators, the latter to difΒferential operators. We will first discuss the construcΒtion of these graphs, and then introduce prototypicaldifference and differential operators, principally realiΒsations of the Laplacian, on each.
But our second goal is to highlight some of theparallels between the two kinds of graphs: indeed,one speaks of Laplacians in both the discrete and themetric case, nomenclature which is justified for varΒious reasons, as we shall see. Finally, we will turnto quantum graphs, which in simple terms are metricgraphs equipped with differential operators. We willdescribe a number of areas of current interest, espeΒcially within (parts of ) themathematical physics comΒmunity. The list of topics we have selected is someΒwhat idiosyncratic; we include a brief mention of, andreferences to the literature for, a variety of others. Thereader interested in discoveringmore is referred to thebook [BK13], considered a standard reference in thearea, the recent survey paper [BK20], the elementaryintroduction [Ber17], and the somewhat older volume[EKKST08], which contains a large number of stilluseful review articles.
1
We give a gentle introduction to analysis on graphs. We focus on the construction of prototypical difference operators on dis-crete graphs, differential operators on metric graphs, and the parallels between the two. The latter lead naturally to quantum graphs, metric graphs on which a SchrΓΆdinger-type differential operator acts, for which we finish by discussing a number of recent applications and ongoing areas of investigation. These are drawn mostly, but not exclusively, from mathematical physics.
CIM Bulletin December 2020.42 29
![Page 30: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/30.jpg)
1.1 DISCRETE GRAPHS
In the case of discrete graphs we are more interestedin the vertices and consider the edges as relations beΒtween the vertices, without necessarily having anydirect physical meaning. More formally, a discretegraph π¦π¦ is a pair (π΅π΅π΅ π΅π΅π΅, where the vertex set π΅π΅ isany countable (in practice usually finite) set and eachedge πΎπΎ in the edge set π΅π΅ may be regarded as a pair ofvertices, that is, π΅π΅ may be identified with a subset ofπ΅π΅π΅π΅π΅. Already herewehave a further decision tomake:whether to treat the edges πΎπΎ πΎ (πΎπΎπ΅ πΎπΎπ΅, πΎπΎπ΅ πΎπΎ π π΅π΅ asordered or unordered pairs; we speak of directed edges(also called bonds in some circles) and undirected edges,respectively. In the case of directed edges πΎπΎ πΎ (πΎπΎπ΅ πΎπΎπ΅,wemay distinguish between the initial vertex πΎπΎ and theterminal vertex πΎπΎ.
Many social networks may be modelled in thisframework; for example, Facebook is a network inwhich each person (or entity) represents a vertex,and being (Facebook) friends corresponds to an undiΒrected edge between the two vertices. Twitter, on theother hand, is directed, if one considers the edge (πΎπΎπ΅ πΎπΎπ΅to mean πΎπΎ is a follower of πΎπΎ β as is the internet itselfwith links being edges between the pages representedby vertices. More generally, any model of a networkin which there is no natural distance between vertices,nor physical bond linking them, is likely to fit into theframework of discrete graphs. This is of course a conΒsiderable simplification; for example, one may assigna weight function to the edges of a discrete graph togive a notion of the proximity of the respective verΒtices.
To do any sort of analysis, of course we need todefine functions on our graph. In the case of discretegraphs, this is easy: if functions live on the vertices,then the space of all functions may be identified withβ|π΅π΅| or β|π΅π΅|. Some care must be taken if the vertexset π΅π΅ is infinite; it becomes natural to work with βππΒspaces.
1.2 METRIC GRAPHS
Metric graphs, on the other hand, focus attention onthe edges, and are thus more suited to modellingactual physical networks, or fine ramified structuressuch as nanostructures. We will write π’π’ πΎ (π’π’ π΅ π’π΅for a metric graph, where now each edge ππ π π’ isidentified with a closed interval which may be finite,of some given length β(πππ΅ π π, i.e. ππ π πππ΅ β(πππ΅π π β,or a halfΒline, πππ΅ +βπ΅. Care must obviously be taken
with the latter, so here we will restrict ourselves tocompact intervals.
In order to encode the topological structure of thegraph, or equivalently to create ametric, one identifiesall interval endpoints which correspond to a given verΒtex. While intuitively this is very simple, formally itis somewhat fiddly and may be done in a number ofways: for example:
β’ identify equivalence classes of endpoints, or
β’ define the underlying metric directly by declarΒing that the distance between two different inΒterval endpoints corresponding to the same verΒtex is zero, thus allowing the construction pathsbetween any two points on different edges, oralternatively
β’ work directly at the level of continuous funcΒtions.
For more details we refer to [BK13, Section 1.3],[Mug19] and [KKLM20, Section 2].
At any rate, this gives rise naturally to a metricspace; the distance between two given points is the(Euclidean) distance of the shortest path betweenthem. Technically the metric is a pseudometric, as itmay take the value +β if there is no path between agiven pair of points, but it becomes a metric if andonly if the graph is connected.
When it comes to defining spaces of functions,metric graphs are, unsurprisingly, more interestingthan their discrete counterparts, albeit not yet at thelevel of πΏπΏππΒspaces: we may simply define, for a graphπ’π’ πΎ (π’π’ π΅ π’π΅ with a finite edge set π’ ,
πΏπΏππ(π’π’ π΅ πΎ β¨ππππ’
πΏπΏππ(πππ΅ π β¨ππππ’
πΏπΏππ(ππ΅ β(πππ΅π΅
(each edge being prototypically equipped withLebesgue measure on the interval πππ΅ β(πππ΅π); indeed,πΏπΏππΒfunctions will never see the vertices as the latterform a set of measure zero. Correspondingly, to inΒtegrate a function over the graph we integrate overeach edge and sum the result. The structure of thegraph is only encoded at the level of continuous funcΒtions: πΆπΆ(π’π’ π΅ will consist of those functions which arecontinuous on every edge, such that their values at allendpointsmeeting at a vertex should agree. These areof course exactly the functions which are continuouswith respect to the metric.
To define differentiable functions becomes morechallenging because of the issue of defining the derivaΒtive across the vertices; instead, it becomes more natΒ
230
![Page 31: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/31.jpg)
[1] This condition is sometimes loosely called a flow in equals flow out condition, although this expression must obviously be interpreted with care, depending on the kind of flow one is imagining.
π£π£1
π£π£2
π£π£3
π£π£4 π£π£5
ππ1
ππ2
ππ3 ππ4
ππππ£π£1)
ππππ£π£2)
ππππ£π£3)
ππππ£π£4) ππππ£π£5)
ππ1
ππ2
ππ3 ππ4 π₯π₯π¦π¦
[0, β1]
[0, β4][0, β3]
[0, β2]
Figure 1: A simple graph on 5 vertices and 4 edges (left); to define a function ππ on a discrete realisation of the graphwe specify the values at the vertices β this also gives rise naturally to difference operators (centre); one may insteadidentify each edge ππππ with an interval [0, βππ] β β (or a halfΒline [0, +β)), and βglue togetherβ the intervals at theirendpoints in the right way, to form a metric graph (right). Here a path between two points π₯π₯ and π¦π¦ is marked inred.
X
X
Figure 2: The Cheeger cut of a graph with 20 vertices.
1 INTRODUCTION: TWO COMPLEMENTARYTYPES OF GRAPHS
Probably everyone has at least an intuitive notion ofwhat a graph is: a collection of vertices, or nodes,joined by edges. Most mathematicians, perhaps evensome nonΒmathematicians, probably have some ideaof the role that graphs play in modelling phenomenaas diverse as fine structures such as crystals and carΒbon nanostructures, social networks, the PageRankalgorithm, data processing and machine learning, β¦,but may not be so familiar with the details.
Generally speaking, at amathematical level, we areinterested in some process taking place on the graph,such as described by a difference or differential equaΒtion. The mathematics behind such equations comΒbines ideas from graph theory (obviously), linear alΒgebra, functional analysis and the theory of differΒential equations, operator theory, and mathematicalphysics; yet many of the details seem to be largely unΒknown to the wider mathematical community. As atest: do you know what quantum graphs are?
Our goal here is to give somewhat uneven introΒduction to analysis on graphs: we first describe, inhopefully accessible terms, what this is: how to deΒfine functions and difference and differential operaΒ
tors on graphs, and study them β and in particularwhat are quantum graphs. Our starting point is thatthere are (at least) two natural, somewhat parallel, noΒtions of graphs: discrete and metric graphs; the forΒmer give rise to difference operators, the latter to difΒferential operators. We will first discuss the construcΒtion of these graphs, and then introduce prototypicaldifference and differential operators, principally realiΒsations of the Laplacian, on each.
But our second goal is to highlight some of theparallels between the two kinds of graphs: indeed,one speaks of Laplacians in both the discrete and themetric case, nomenclature which is justified for varΒious reasons, as we shall see. Finally, we will turnto quantum graphs, which in simple terms are metricgraphs equipped with differential operators. We willdescribe a number of areas of current interest, espeΒcially within (parts of ) themathematical physics comΒmunity. The list of topics we have selected is someΒwhat idiosyncratic; we include a brief mention of, andreferences to the literature for, a variety of others. Thereader interested in discoveringmore is referred to thebook [BK13], considered a standard reference in thearea, the recent survey paper [BK20], the elementaryintroduction [Ber17], and the somewhat older volume[EKKST08], which contains a large number of stilluseful review articles.
1
Figure 1. A simple graph on 5 vertices and 4 edges (left); to define a function f on a discrete realisation of the graph we specify the values at the vertices β this also gives rise naturally to difference operators (centre); one may instead identify each edge ei with an interval [0,πi] β β$ (or a half-line [0,+β)), and glue together the intervals at their endpoints in the right way, to form a metric graph (right). Here a path between two points x and y is marked in red.
ural to speak of vertex conditions which the functionsshould satisfy (such as continuity at the vertices, asimposed in πΆπΆπΆπΆπΆ πΆ). Two of the most natural suchconditions to satisfy are the Dirichlet, or zero, condiΒtion, and the Kirchhoff condition, where the sum ofinwardΒpointing derivatives at a vertex equals zero.[1]
In practice, one usually works with Sobolevspaces of weakly differentiable functions; for examΒple, π»π»1πΆπΆπΆ πΆ is defined as those functions which areedgewise πΏπΏ2Βintegrable with edgewise πΏπΏ2Βintegrableweak derivative, and which are continuous across thevertices. This makes sense since by standard Sobolevembedding theorems oneΒdimensional π»π»1Βfunctionsare continuous and thus, up to choosing the correctrepresentative, defined pointwise.
Three final observations are in order: firstly, thusdefined, our graphs are not considered to be embedΒded in Euclidean space; there is no curvature of theedges or angle between them. Secondly, by labellingone vertex of an edge ππ π πππ ππΆπππΆπ as π and the otheras ππΆπππΆ π π, we are implicitly (or explicitly) imposingan orientation. However, for many practical purposesthis orientation is irrelevant; the differential operatorswe shall define in the sequel are independent of thischoice up to unitary equivalence. Thirdly, in the caseof metric graphs it is easy to allow multiple edges beΒtween vertices, as well as loops (edges which beginand end at the same vertex); in the case of discretegraphs this is a bit more complicated and we will tacΒitly assume that our graphs are free of such features,even thoughmost of what we discuss will remain trueeven with multiple parallel edges and loops.
2 DIFFERENCE AND DIFFERENTIALOPERATORS
We see immediately that on discrete graphs, since thefunctions are identifiable with vectors, difference opΒerators (or more generally matrices) will arise; whileon metric graphs we may define (ordinary) differenΒtial expressions on the edges. In the latter case thepoint of interest becomes specifying the vertex condiΒtions, or equivalently the domain of definition of thedifferential operator; a metric graph is essentially asmooth oneΒdimensional manifold with isolated sinΒgularities (the vertices). In both cases we will illusΒtrate this via a prototypical operator, the Laplacian;note that here, in both cases, our edges will be undiΒrected.
Let us start with metric graphs, as here we arecloser to the traditional Laplacian from the theory ofPDEs. In fact, we start with the differential expresΒsion βππ β³ on each edge. It is natural to impose contiΒnuity at all vertices, as this is essentially the minimalrequirement for the functions to see the graph. AddiΒtionally imposing the Kirchhoff condition, which wemay write as
βππ adjacent to π£π£
ππππππππ
πΆπ£π£πΆ π£ ππ
that is, the sum of the derivatives of ππ at the endpointof each edge ππ directed into the vertex π£π£, gives rise tothe Laplacian with vertex conditions variously knownas standard, natural, continuityΒKirchhoff, and evenNeumannΒKirchhoff (if the vertex has degree one, i.e.,
3CIM Bulletin December 2020.42 31
![Page 32: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/32.jpg)
only one edge attached, then this reduces to the NeuΒmann condition. Roughly speaking, in many waysthe Laplacianwith standard conditions behaves someΒwhat like the Neumann Laplacian on domains, orthe LaplaceΒBeltrami operator on manifolds withoutboundary). For the Dirichlet condition at a vertexπ£π£, instead of the Kirchhoff condition we require thatππππ£π£π π π.
If the graph π’π’ has finite total length, then suchoperators are selfΒadjoint, semiΒbounded from below,and have compact resolvent; thus they behave exactlylike Laplacians or SchrΓΆdinger operators on boundeddomains and manifolds. Generalisations, such asadding a potential to each edge, are easy to incorpoΒrate in this framework.
All this is perhaps more naturally seen at thelevel of forms/weak solutions: the associated positive,symmetric sesquilinear form reads
πππππ π πππ π β«π’π’ππ β² β ππβ² dπ₯π₯
with form domain exactly π»π»1ππ’π’ π in the case of thestandard Laplacian; if Dirichlet conditions are imΒposed at one or more vertices then the functionsshould additionally take on the value π there. (Allthis is a short exercise in integration by parts.) Theeigenvalues and eigenfunctions of the Laplacian adΒmit the usual minΒmax variational characterisation;for example, the smallest eigenvalue can be obtainedby minimising πππππ π ππ π among all ππ π π»π»1ππ’π’ π whoseπΏπΏ2Βnorm is 1.
On discrete graphs, the (discrete or combinatorial)Laplacian is defined purely in terms of the graph strucΒture. We suppose π¦π¦ π ππ¦π¦π π¦π¦π to be a discrete graphwith finite vertex set π¦π¦ π π΅π΅π΅1π β¦ π π΅π΅ππ} and finite edgeset π¦π¦ π π΅π€π€1π β¦ π π€π€ππ}. We take as a starting point thefollowing matrices:
β’ the adjacency matrix, the symmetric matrixwhose ππππ πππΒentry is 1 if π΅π΅ππ and π΅π΅ππ share an edge,or π otherwise (in the case of directed edges thismatrix can still be defined but will no longer besymmetric);
β’ the degree matrix, the diagonal matrix whoseππππ πππΒentry is the degree of π΅π΅ππ, i.e., the numberof edges emanating from π΅π΅ππ.
The (discrete) Laplacian is the difference operator corΒresponding to the symmetric, positive semidefinitematrix πΏπΏ πΏπ πΏπΏ πΏ πΏπΏ. For example, for the graph deΒpicted in Figure 1, with the order of vertices as speciΒ
fied there, the Laplacian would be
πΏπΏ π
ββββββ
1 πΏ1 π π ππΏ1 3 πΏ1 πΏ1 ππ πΏ1 1 π ππ πΏ1 π 2 πΏ1π π π πΏ1 1
ββββββ
.
The fact that this is a plausible discrete version of theLaplacian may be recognised in (at least) two ways:
β’ vectors π₯π₯ satisfying πΏπΏπ₯π₯ π π have the mean valueproperty, as can be checked with the above examΒple: since the sum of each row of πΏπΏ is zero, thevalue of π₯π₯ at a vertex is equal to the sum of thevalues at the surrounding vertices β just as harΒmonic functions inβππ , solutions ofΞππ π π, satΒisfy the (continuous) mean value property;
β’ at the level of forms: πΏπΏ is associated with thepositive, symmetric sesquilinear form
ππππ₯π₯π πππ π βπ€π€ππ¦π¦
πβ ππ π₯π₯πππ€π€ππβ ππ πππππ€π€ππ
where β π βπππππ is the soΒcalled (signed) inci-dence matrix encoding which vertices are the terΒminal and initial endpoints of which edges; infact πΏπΏ may also be represented as πΏπΏ π β β ππ ,and we may intuitively think of β as a discretecounterpart of the divergence operator.
A word of caution: there is a common, normalisedvariant, namely
πΏπΏnorm πΏπ Id πΏπΏπΏπΏ1/2πΏπΏπΏπΏπΏ1/2
(that is, we normalise the operator by the degree ofthe vertices); its |π¦π¦| eigenvalues (counting multiplicΒities) always lie in the interval [ππ 2]. The standardreference on the topic is [Chu97]; see also [Mug14,Chapter 2] for the construction of all these matricesas well as their directed counterparts.
A strong mathematical parallel between the stanΒdard Laplacian on a metric graph and the normalisedLaplacian on the corresponding discrete graph wasestablished by von Below in 1985 [Bel85]. Namely,if all the edges of the metric graph have length 1,and we denote by ππππ the ordered eigenvalues of thestandard Laplacian, then, up to certain special cases(corresponding to the normalised Laplacian eigenvalΒues π and 2) the eigenvalues of πΏπΏnorm are given by1 πΏ cosπβπππππ; the values of the eigenvectors correΒspond to the values of the eigenfunctions at the verΒtices. Thus, at least for this special class of equilateralmetric graphs, the standard Laplacian as a differentialoperator is essentially determined by the corresponding
4
32
![Page 33: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/33.jpg)
discrete (normalised) Laplacian. Such connectionsbetween discrete and continuous versions have natuΒrally been explored further over the last 30Βodd years;see [LP16] and the references therein.
2.1 SO WHAT ARE QUANTUM GRAPHS?
We can now finally answer the first question posed inthe title. By a quantum graph we understand a metΒric graph on which acts a differential operator, mostoften (but not necessarily) some kind of SchrΓΆdingeroperator orHamiltonian [Ber17, BK13]; of course thisincludes various realisations of the Laplacian.
While differential operators onmetric graphs havebeen studied for a long timeβ theywere actively studΒied in the 1980s, often under the name ππ2-networks(e.g., [Bel85, Nic87]), and there are applications goΒing back much further [RS53] β the name quantumgraph is generally considered [Ber17] to trace back tothe article Quantum Chaos on Graphs [KS97] from1997, possibly as a contraction of the title.
3 A CORNUCOPIA OF APPLICATIONS
We finish with a discussion of some current topics ofinterest in the community, as an answer to the secΒond question posed in the title: we wish to give someidea of the variety of problems and applications whicharise in the context of quantum graphs. One maybroadly and imperfectly group the applications intomodels where it is intrinsically sensible to considerramified structures (atomic or crystalline structures,honeycombs, ramified traps,β¦) and those where thegraph represents a toymodel used to studymathematΒical or quantum physical phenomena: graphs are simΒple oneΒdimensional objects which often display comΒplex behaviour typical of higherΒdimensional probΒlems.
In the following list no claim is made to completeΒness, either in the list of topics or in the referencesgiven, which largely reflect the authorβs personal tasteand prejudices. Where possible we have tried to proΒvide some of the most recent references available toact as a starting point for a further literature search.
In keepingwith these prejudices, aswell as the genΒeral focus of the quantum graph community, we willmostly be interested in differential operators such asthe Laplacian and SchrΓΆdinger operators, and theirspectra. This is natural since by the spectral theorem
the spectrum completely determines such selfΒadjointoperators.
3.1 APPROXIMATION OF, OR BY,HIGHERΒDIMENSIONAL OBJECTS
There are two senses in which graphs, be they disΒcrete or metric, can be related to higherΒdimensionaldomains or manifolds: one can consider a (metric)graph as the limit of a sequence of thin branching doΒmains (shrinking tubes, or fattened graphs), or one cantry and approximate a domain ormanifold as the limitof as sequence of graphs. In the latter case one usuallytakes discrete graphs as the approximating objects, asa kind of discretisation of the domain or manifold.
Needless to say, there is an extensive literature onboth. The latter is sometimes used to extend resultsfrom discrete graphs to manifolds (as in [LLPO15],see also Section 3.3). The former provides a justificaΒtion for using quantum graphs to study phenomenalike waveguides, be they acoustic, quantum or elecΒtromagnetic, thin superΒconducting structures and soon; here we will follow, and refer to, [BK13, SecΒtion 7.5]. Another standard reference for shrinkingtubes is the review paper [Gri08] contained in the volΒume [EKKST08]. Typical questions include whetherthe solutions of differential equations in the thin doΒmains converge to the solution of some differentialequation on the graph, and if so, what vertex condiΒtions the problem in the limit satisfies. (More techΒnically, we are interested in convergence of the resolΒvents of the operators in various norms, as well as ofthe operator eigenvalues and eigenfunctions.)
The precise results depend very much on the naΒture of the approximation, but in perhaps the simΒplest and most important case of SchrΓΆdinger operaΒtors inNeumann tubes (thin perfectly insulated tubes)shrinking uniformly, one does at least have converΒgence of the eigenvalues to the eigenvalues, in the corΒrect order, of the SchrΓΆdinger operator on the graphwith standard vertex conditions, and where the elecΒtric potential is, roughly speaking, the restriction ofthe potential on the thin domain to the graph it conΒtains. Work is still ongoing to establish other kindsof convergence, in particular under different kinds ofdomain convergence.
In this case the limit object, the quantum graphwith its SchrΓΆdinger operator, forgets many geometΒric features of the domains, such as angles betweenbranches, curvature of edges and so on. If one alΒlows the Neumann tubes to shrink in a wilder, nonΒ
5CIM Bulletin December 2020.42 33
![Page 34: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/34.jpg)
uniform fashion, then one may obtain more interestΒing limit quantum graphs, including where the opΒerators satisfy other vertex conditions than the stanΒdard/Kirchhoff ones.
3.2 SPECTRAL GEOMETRY
We have seen that discrete Laplacians may be defineddirectly in terms of the structure (topology) of a disΒcrete graph, and that at least at the spectral level thiscan be transferred to equilateral metric graphs via vonBelowβs formula. Can we say something about (nonΒequilateral) metric graphs, where we have to contendwith both the topology and the edge length?
In the case of domains and manifolds a groupof questions revolves around understanding how theeigenvalues and eigenfunctions depend on the geomΒetry of the underlying domain or manifold. The clasΒsical example is the theorem of FaberΒKrahn from the1920s, based on an earlier conjecture of LordRayleigh,that among all domains of given volume the ball isthe one whose first Dirichlet Laplacian eigenvalue issmallest. This is an analytic translation of the geometΒric isoperimetric inequality, that the ball minimisessurface area for given volume; the first (nonzero)eigenvalue is of particular interest because it controlsthe rate of heat loss in the heat equation, the lowest freΒquency of the object, and so on. We refer to [Pay67]for a (classical) introduction and [Hen06, Hen17] formore modern surveys of the area of shape optimisationand spectral theory. The corresponding inverse probΒlem, determining the domain/manifold based on thespectrum of a differential operator on it, correspondsto the question made famous by Mark Kac, βcan onehear the shape of a drum?β; see [LR15].
On metric graphs the equivalent of the theoremof FaberΒKrahn states that the smallest nonzero eigenΒvalue of the Laplacianwith standard vertex conditionsisminimisedwhen the graph is an interval of the samelength; this theorem first appeared around 30 yearsago [Nic87]. It turns out that graphs are far moreamenable to this kind of analysis than domains; see[BL17, BKKM19, KKMM16]. A surprisingly subtlequestion is which (geometric or topological) properΒties of a graph are sufficient to bound its eigenvaluesand which are not. For example, fixing the diameΒter π·π· (length of the longest path within the graph)alone places no control on the smallest nonzero stanΒdard Laplacian eigenvalue: it may be arbitrarily largeor small [KKMM16]; however, if we restrict to trees,graphs without cycles, then it cannot exceed ππ2/π·π·2,
the corresponding eigenvalue of an interval of lengthπ·π·.
Work has also been done on isospectral graphs,quantum graphs which are different but have thesame Laplacian spectra. On graphs the problem canbe given a new twist since one has more chance of deΒscribing the corresponding eigenfunctions: one conΒsiders the soΒcalled nodal count, the number of nodaldomains, which are by definition the connected comΒponents of the set where the eigenfunction is nonzero.See [BK13, Section 7.1].
3.3 CLUSTERING AND PARTITIONS
A major preoccupation in applied graph theory is todetect the presence of clusters in a (usually discrete)graph. One might ask whether a given social networksuch as Facebook tends to be divided into groups ofhig hly interconnected individuals with few links beΒtween the groups, thus creating the infamous echochambers. Alternatively, one might wish to identify,say, weaknesses in a road network or an electricitygrid: if the power lines here go down, does half thecountry lose power?
There are various ways to measure this. One natuΒral way is the notion of Cheeger constants and Cheegercuts borrowed from geometric analysis, originally inΒtroduced for manifolds. Say we wish to cut the graphπ¦π¦ into two pieces ππ and ππππ = π¦π¦ π¦ ππ, which we do bycutting through edges. Then for each possible cut welook at the ratio
|ππππ|min{|ππ|π |ππππ|}
of edges cut |ππππ| to the smaller of the two sets ππ e ππππ,as measured by the number of vertices in the set.
The infimum of this quotient over all possible cutsis the Cheeger constant; the smaller the constant, theeasier it is to cut the graph into two (the traditionalimage for this is the dumbbell manifold, cut throughits thin handle).
Figure 2 gives an example on graphs: on this graphof 20 vertices, there is a way to make just two cutsto separate the graph into two groups of 10 verticeseach, labelled as blue and red; this is in fact the optiΒmal cut. One might imagine a social network wherethe vertices represent users; the blue users tend onlyto have friends with other blue users, while the redusers likewise stay amongst themselves. In this casethe Cheeger constant will be 2/ min{10π 10} = 1/5,which may be considered small (the number has noabsolute meaning but should be viewed in conjuncΒ
634
![Page 35: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/35.jpg)
tion with the total number of vertices and edges).
One can also consider higher order Cheeger conΒstants, which partition the graph into more than twopieces, and in fact estimating these constants, in parΒticular in terms of Laplacian eigenvalues, is one inΒstance where results were first proved on discretegraphs and then transferred to manifolds [LLPO15].Cheeger constants have also been introduced on metΒric graphs, with the numerator remaining the sameand the denominator becoming the total length (sumof edge lengths) of each piece [KM16, Nic87].
The Cheeger constant corresponds to the firsteigenvalue of the soΒcalled 1ΒLaplacian, i.e., the ππΒLaplacian when ππ π 1; the eigenvector/eigenfunctionhas two nodal domains which correspond to theCheeger cut. This operator is singular, its eigenvalΒues lack the easyπΏπΏππΒvariational characterisation of theππΒLaplacian eigenvalues, and actually calculating theconstant of a large graph becomes a computationallyhard problem.
Thus one can use the (2Β)Laplacian (say, withstandard vertex conditions) as a natural proxy, as itsvariational structure makes determining the eigenvalΒues and eigenfunctions much easier, both analyticallyand computationally. Ideally one would use the nodaldomains of the ππΒth eigenvalue as an βoptimalβ partiΒtion into ππ pieces, but in general there is no simple reΒlationship between the number of nodal domains ofan eigenfunction and its number in the sequence. Anatural alternative is to consider spectral minimal par-titions, whereby one looks to minimise a functional ofthe eigenvalues over all partitions; prototypically thisproblem might take the form
infπ«π«
maxπππ1πππππ
ππ1(Ξ©ππ)π
where the infimum is taken over all partitionsπ«π« of theobject (domain, graph, β¦) into ππ pieces Ξ©1π π π Ξ©ππ,and ππ1(Ξ©ππ) is the first nontrivial eigenvalue of a suitΒable Laplacian onΞ©ππ; one could equally take a ππΒnormof the eigenvalues in place of theβΒnorm. Such probΒ
lems were originally considered, and have been studΒied intensively, on domains and some manifolds; see[Hen17, Chapter 10] for a survey.
On metric graphs this topic is new: the first sysΒtematic study of spectral minimal partitions was unΒdertaken in [KKLM20]. As is the case for spectral geΒometry, and actually for many problems consideredhere, one can say far more on metric graphs than ondomains. Here, far more functionals can be meanΒingfully defined on the former than the latter, includΒing more exotic combinations of eigenvalues (suchas maxΒmin rather than minΒmax problems). UnderΒstanding how these optimal partitions differ andwhatthey reveal about the structure of the graph will be atopic of interest in the next few years.
3.4 NONLINEAR SCHRΓDINGER EQUATIONS
Until now we have always considered linear differΒential operators, as has historically usually been thecase on metric graphs. There is, however, a notablefamily of exceptions, first considered just a few yearsago [AST15a]. This principally involves studying exΒistence, or nonexistence, of certain solutions of staΒtionary nonlinear SchrΓΆdinger equations on metricgraphs (NLSE for short). A stationaryNLSE typicallytakes the form
βΞπ’π’ π’ π’π’(π’π’) π πππ’π’π (1)
where in place of the usual potential term ππ π’π’ a nonΒlinearity π’π’(π’π’) is introduced; here, as in the literature,we will consider the prototypical power nonlinearityπ’π’(π’π’) π ππ’π’πππβ1π’π’. These equations, a bedrock of theCalculus of Variations literature, are most commonlystudied in ππΒdimensional space, see [Caz03] for anintroduction, but a number of applications, such asBoseΒEinstein condensates in traps or optical fibres[AST15a], make it reasonable to consider NLSE inramified structures, that is, on metric graphs, mostcommonly and naturally with standard vertex condiΒ
7
π£π£1
π£π£2
π£π£3
π£π£4 π£π£5
ππ1
ππ2
ππ3 ππ4
ππππ£π£1)
ππππ£π£2)
ππππ£π£3)
ππππ£π£4) ππππ£π£5)
ππ1
ππ2
ππ3 ππ4 π₯π₯π¦π¦
[0, β1]
[0, β4][0, β3]
[0, β2]
Figure 1: A simple graph on 5 vertices and 4 edges (left); to define a function ππ on a discrete realisation of the graphwe specify the values at the vertices β this also gives rise naturally to difference operators (centre); one may insteadidentify each edge ππππ with an interval [0, βππ] β β (or a halfΒline [0, +β)), and βglue togetherβ the intervals at theirendpoints in the right way, to form a metric graph (right). Here a path between two points π₯π₯ and π¦π¦ is marked inred.
X
X
Figure 2: The Cheeger cut of a graph with 20 vertices.
1 INTRODUCTION: TWO COMPLEMENTARYTYPES OF GRAPHS
Probably everyone has at least an intuitive notion ofwhat a graph is: a collection of vertices, or nodes,joined by edges. Most mathematicians, perhaps evensome nonΒmathematicians, probably have some ideaof the role that graphs play in modelling phenomenaas diverse as fine structures such as crystals and carΒbon nanostructures, social networks, the PageRankalgorithm, data processing and machine learning, β¦,but may not be so familiar with the details.
Generally speaking, at amathematical level, we areinterested in some process taking place on the graph,such as described by a difference or differential equaΒtion. The mathematics behind such equations comΒbines ideas from graph theory (obviously), linear alΒgebra, functional analysis and the theory of differΒential equations, operator theory, and mathematicalphysics; yet many of the details seem to be largely unΒknown to the wider mathematical community. As atest: do you know what quantum graphs are?
Our goal here is to give somewhat uneven introΒduction to analysis on graphs: we first describe, inhopefully accessible terms, what this is: how to deΒfine functions and difference and differential operaΒ
tors on graphs, and study them β and in particularwhat are quantum graphs. Our starting point is thatthere are (at least) two natural, somewhat parallel, noΒtions of graphs: discrete and metric graphs; the forΒmer give rise to difference operators, the latter to difΒferential operators. We will first discuss the construcΒtion of these graphs, and then introduce prototypicaldifference and differential operators, principally realiΒsations of the Laplacian, on each.
But our second goal is to highlight some of theparallels between the two kinds of graphs: indeed,one speaks of Laplacians in both the discrete and themetric case, nomenclature which is justified for varΒious reasons, as we shall see. Finally, we will turnto quantum graphs, which in simple terms are metricgraphs equipped with differential operators. We willdescribe a number of areas of current interest, espeΒcially within (parts of ) themathematical physics comΒmunity. The list of topics we have selected is someΒwhat idiosyncratic; we include a brief mention of, andreferences to the literature for, a variety of others. Thereader interested in discoveringmore is referred to thebook [BK13], considered a standard reference in thearea, the recent survey paper [BK20], the elementaryintroduction [Ber17], and the somewhat older volume[EKKST08], which contains a large number of stilluseful review articles.
1
Figure 2. The Cheeger cut of a graph whith 20 vertices.
CIM Bulletin December 2020.42 35
![Page 36: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/36.jpg)
tions. Ofmost interest are the ground states, minimisΒers of the energy functional for which 1 is the EulerΒLagrange equation:
πΈπΈπΈπΈπΈπΈ πΈ 12
βπΈπΈβ²β22 β 1
ππβπΈπΈβππ
ππ,
where β β β2 and β β βππ are, respectively, the πΏπΏ2Β andπΏπΏππΒnorms, here on some graph π’π’ . Here one usuΒally considers unbounded graphs, with a finite numΒber of edges but where some of them are halfΒlines(π’π’ πΈ β itself is a prototype, being two halfΒlinesglued together at the origin), as well as the subcriticalcase 2 < ππ < π, which guarantees the Sobolev embedΒding π»π»1 βͺ πΏπΏππ in dimension 1.
It turns out that the existence or nonΒexistence ofground states on such graphs depends heavily on thetopology of the graph, as shown in a series of landΒmark papers [AST15a, AST15b, AST16, AST17]. FurΒther research, including into stability of solutions andstanding waves, other types of metric graphs, otherrestrictions on the parameters, and other equationsis ongoing; see, for example, [DST20, Hof19, NP20]and the references therein.
3.5 FINAL REMARKS
The above list excludes a huge and growing numΒber of topics from various areas of mathematics.We could mention quantum chaos (the presumablesource of the name quantum graph, as discussed inSection 2.1; see also [BK13, Chapter 6]), as well asvarious other applications in mathematical physicssuch as scattering and inverse scattering, the BetheΒSommerfeld property on the gap structure of the specΒtrum of periodic objects [ET17], Anderson localisaΒtion [DFS, DS19], the spectra of graphene and carΒbon nanotubes, BoseΒEinstein condensates, and thequantumHall effect. Differential equations on metricgraphs also feature in other areas ofmathematics as diΒverse as neural networks andmodels of population dyΒnamics [DLPZ20, SCA14]. Surveys of many of theseand further applications in mathematical physics maybe found in [BK20], [BK13, Chapter 7] and the collecΒtion [EKKST08].
REFERENCES
[AST15a] R. Adami, E. Serra and P. Tilli, Lack ofground state for NLSE on bridge type graphs,Mathematical Technology of Networks, 1β11,
Springer Proc. Math. Stat., vol. 128, Springer,Cham, 2015.
[AST15b] R. Adami, E. Serra and P. Tilli,NLS groundstates on graphs, Calc. Var. Partial DifferentialEquations 54 (2015), 743β761.
[AST16] R. Adami, E. Serra and P. Tilli, Thresholdphenomena and existence results for NLS groundstates on metric graphs, J. Funct. Anal. 271(2016), 201β223.
[AST17] R. Adami, E. Serra and P. Tilli, Negative en-ergy ground states for the πΏπΏ2-critical NLSE onmetric graphs, Comm.Math. Phys. 352 (2017),387β406.
[BL17] R. Band and G. LΓ©vy, Quantum graphs whichoptimize the spectral gap, Ann. Henri PoincarΓ©18 (2017), 3269β3323.
[Bel85] J. von Below, A characteristic equation associ-ated to an eigenvalue problem on ππ2-networks,Linear Algebra Appl. 71 (1985), 309β325.
[Ber17] G. Berkolaiko, An elementary introduction toquantum graphs, Geometric and computaΒtional spectral theory 700 (2017), 41β72.
[BKKM19] G. Berkolaiko, J. B. Kennedy, P. KurasovandD.Mugnolo, Surgery principles for the spec-tral analysis of quantum graphs, Trans. Amer.Math. Soc. 372 (2019), 5153β5197.
[BK13] G. Berkolaiko and P. Kuchment, Introductionto quantum graphs, Mathematical Surveys andMonographs, vol. 186, American MathematiΒcal Society, Providence, RI, 2013.
[BK20] J. Bolte and J. Kerner,Many-particle quantumgraphs: a review, pp. 29β66 in F. M. Atay etal (eds.), Discrete and Continuous Models inthe Theory of Networks, Operator Theory:Advances and Applications 281, BirkβοΏ½hauser,Cham, 2020.
[Caz03] T. Cazenave, Semilinear SchrΓΆdinger equa-tions, Courant Lecture Notes in MathematΒics, vol. 10, American Mathematical Society,Providence, RI, 2003.
[Chu97] F.R.K. Chung, Spectral graph theory, CBMSRegional Conference Series in MathematΒics, vol. 92, American Mathematical Society,Providence, RI, 1997.
8
36
![Page 37: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/37.jpg)
[DFS] D. Damanik, J. Fillman and S. Sukhtaiev, Lo-calization for Anderson models on metric and dis-crete tree graphs, Math. Ann. 376 (2020), 1337β1393.
[DS19] D. Damanik, S. Sukhtaiev, Anderson localiza-tion for radial tree graphs with random branchingnumbers, J. Funct. Anal. 277 (2019), 418β433.
[DST20] S. Dovetta, E. Serra and P. Tilli, Unique-ness and non-uniqueness of prescribed mass NLSground states on metric graphs, Adv. Math. 374(2020), 107352, 41 pp.
[DLPZ20] Y. Du, B. Lou, R. Peng and M. Zhou, TheFisher-KPP equation over simple graphs: variedpersistence states in river networks, J.Math. Biol.80 (2020), 1559β1616.
[EKKST08] P. Exner, J. Keating, P. Kuchment,T. Sunada and A. Teplyaev (eds.), Analysis ongraphs and its applications. Proc. Sympos. PureMath., vol. 77, American Mathematical SociΒety, Providence, RI, 2008.
[ET17] P. Exner and O. Turek, Periodic quantumgraphs from the Bethe-Sommerfeld perspective, J.Phys. A 50 (2017), 455201, 32 pp.
[Gri08] D. Grieser, Thin tubes in mathematical physics,global analysis and spectral geometry, Analysison graphs and its applications, 565β593, Proc.Sympos. PureMath., vol. 77, AmericanMathΒematical Society, Providence, RI, 2008.
[Hen06] A. Henrot, Extremum problems for eigenval-ues of elliptic operators, Frontiers in MathematΒics, BirkhΓ€userΒVerlag, Basel, 2006.
[Hen17] A. Henrot (ed.), Shape optimization and spec-tral theory, De Gruyter Open, Warsaw, 2017.
[Hof19] M. Hofmann, An existence theory for nonlin-ear equations on metric graphs via energy meth-ods, preprint (2019), arXiv:1909.07856.
[KKLM20] J. B. Kennedy, P. Kurasov, C. LΓ©naand D. Mugnolo, A theory of spectral par-titions of metric graphs, preprint (2020),arXiv:2005.01126.
[KKMM16] J. B. Kennedy, P. Kurasov, G. MalenovΓ‘and D. Mugnolo,On the spectral gap of a quan-tum graph, Ann. Henri PoincarΓ© 17 (2016),2439β2473.
[KM16] J. B Kennedy and D. Mugnolo, The Cheegerconstant of a quantum graph, PAMM Proc.Appl. Math. Mech. 16 (2016), 875β876.
[KS97] T. Kottos and U. Smilansky, Quantum chaoson graphs, Phys. Rev. Lett. 79 (1997), 4794β4797.
[LLPO15] C. Lange, S. Liu, N. Peyerimhoff andO. Post, Frustration index and Cheeger inequal-ities for discrete and continuous magnetic Lapla-cians, Calc. Var. 54 (2015), 4165β4196.
[LP16] D. Lenz and K. Pankrashkin,New relations be-tween discrete and continuous transition opera-tors on (metric) graphs, Integral Equations OpΒerator Theory 84 (2016), 151β181.
[LR15] Z. Lu and J. Rowlett, The sound of symmetry,Amer. Math. Monthly 122 (2015), 815β835.
[Mug14] D. Mugnolo, Semigroup methods for evo-lution equations on networks, UnderstandingComplex Systems, Springer, Cham, 2014.
[Mug19] D. Mugnolo, What is actually a metricgraph?, preprint (2019), arXiv:1912.07549.
[Nic87] S. Nicaise, Spectre des rΓ©seaux topologiques fi-nis, Bull. Sci. Math. (2) 111 (1987), 401β413.
[NP20] D. Noja and D. E. Pelinovsky, Standing wavesof the quintic NLS equation on tadpole graphs,Calc. Var. Partial Differential Equations 59(2020), 173.
[Pay67] L. E. Payne, Isoperimetric inequalities and theirapplications, SIAM Rev. 9 (1967), 453β488.
[RS53] K. Ruedenberg and C. W. Scherr,FreeοΏ½electron network model for conjugatedsystems. I. Theory, J. Chem. Phys. 21 (1953),1565β1581.
[SCA14] J. Sarhad, R. Carlson and C. E. Anderson,Population persistence in river networks, J.Math.Biol. 69 (2014), 401β448.
9CIM Bulletin December 2020.42 37
![Page 38: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/38.jpg)
an interview with
AndrΓ© Neves
* Departamento de MatemΓ‘tica, Faculdade de CiΓͺncias, Univ. de Lisboa, Edf. C6, Campo Grande 1749-016 Lisboa, Portugal [email protected]
by Carlos Florentino*
AndrΓ© Neves is an outstanding portuguese mathematician working in the U.S.A. He has held research and professorship positions at Princeton University, at the Imperial College of London and at the University of Chicago, where he is based now. He obtained his undergraduate degree in 1999, at Instituto Superior TΓ©cnico, in Lisbon, and the Ph.D. in 2005 at Stanford University, under the supervision of Richard Schoen. He works on Differential Geometry and Analysis of Partial Differential Equations. Among several distinctions and awards, he was invited speaker at the International Congress of Mathematicians in Seoul (2014) and was awarded a New Horizons in Mathematics Prize in 2015 for βoutstanding contributions to several areas of differential geometry, including work on scalar curvature, geometric flows, and his solution (with CodÑ· Marques) of the 50-year-old Willmore Conjecture.β He also received the Oswald Veblen Prize in Geometry in 2016 (conferredby the American Mathematical Society), and was recently elected to the American Academy of Arts and Sciences. AndrΓ© Neves was the lecturer of this yearβs Pedro Nunes Lectures, with a seminar entitled: Counting minimal surfaces in negatively curved 3-manifolds, the first of this series which took place online. As in previous editions, we took this opportunity for a short interview.
38
![Page 39: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/39.jpg)
Let us start by talking about your early years. In School/High School, was Mathematics your favorite subject? What other subjects captured your interest?I always loved Mathematics. My fondest memories from High School are of being alone in my room studying Mathematics, while listening to music.
Can you tell us about your decision to study at Instituto Superior TΓ©cnico (IST)? What made you choose the undergraduate degree in Mathematics and Computation?It was non-linear, I am afraid. I first chose to pursue an Engineering degree because I could not imagine that one could be a mathematician. I literally thought that all Mathematics had been done by Cauchy, Bolzano, and Weierstrass. Only when I took an ODEβs class with Professor JosΓ© Sousa Ramos I realized that there was a whole world out there to be explored. I then changed my major from Engineering to Mathematics.
Can you describe the study environment at IST during your time there, in particular the role of your professors and colleagues in your Mathematics background, and in your path to become a researcher?I loved it! It was a very small group and I had the good fortune of being taught by several young mathematicians that had just finished their Ph.D.βs and returned from abroad (mostly from the United States). They would teach Mathematics that seemed very original, sophisticated, and it was truly inspiring.
It is clear that your Ph.D. at Stanford University was a fundamental step in your career. Can you also describe the academic environment there, in particular how you came to work on the interface between Geometry and Analysis, and the role of Prof. Richard Schoen, your advisor?On my first year, I took a course in Riemannian Geometry that was taught by Rick Schoen. I loved the class and the subject and decided to pursue my Ph.D. in that area. Rick was the first outstanding mathematician I met, and he has served as a role model since. He was not afraid of pursuing hard questions and if he felt he had a good idea, he would fiercely pursue it. Most importantly, I understood that there is always a good reason for an idea to work, and that trying things just for the sake of it rarely works. He shaped my mathematical career.
Other key moments in your career were the Postdoc at Princeton, the position at Imperial College and the award of an ERC (European Research Council) grant. Can you tell us about those periods?The time at Princeton was a bit stressful because one of the hardest periods in a mathematics career is the transition from Postdoc to mathematician. There is a tension that
comes from the fact that, on one hand, we have to come up with our own problems and carve our own way of doing Mathematics, but on the other hand we also have the pressure to have papers published, because we will be applying for tenure-track jobs soon. At Imperial College, I had a tenured job and so the pressure of publishing decreased. I was freer to pursue other directions in Mathematics and to tackle problems in which I had to start from scratch. The ERC grant helped me because it reduced my administrative duties and allowed me to hire postdocs, some of which became good collaborators of mine.
You have established many research collaborations, but the one with Fernando CodΓ‘ Marques is noteworthy. Can you summarize the story of how you became collaborators and friends?We met at Princeton and became friends as we were one of the few Portuguese speaking people at the Mathematics Department. We were working on the same field and so we kept discussing mathematics problems on a regular basis. For the first years not much happened, but after a while we were able to look at some old problems with a new point of view and then our mathematics collaborations started.
Some mathematicians follow mostly one problem or guiding principle in their research, others keep changing fields and exploring different topics. Do you have a main philosophical guiding principle?It is hard for me to answer because I literally pursue the questions that interest me at any given time. As I have become more confident, I have learned that if there is some phenomenon I donβt understand, then it is probably worthwhile to pursue that.
You are probably the portuguese mathematician that received more international prizes and awards. What do you think are the most important qualities a researcher must have to achieve such success at the international level?
Being courageous and bold in the sense of not being afraid of addressing problems that are perceived as being hard is a good quality to have, in my opinion.
You have worked both in Europe and in the United States. Do you think there are key differences in the ways Mathematics is viewed by academic departments, and their approaches to research training and funding?In the U.S. the grants tend to be smaller, but more mathematicians are funded. In Europe, the grants are higher but less mathematicians are funded. I think that is partly because the University Departments in the U.S. fund the postdocs and the students, and so less funding is needed at the individual level. That being said, I am not sure that has
CIM Bulletin December 2020.42 39
![Page 40: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/40.jpg)
any effect in the quality of research. Both continents have stellar mathematicians.
Even though the World today is facing many crisis, and the areas of research and development are constantly evolving, many European countries have recently invested in Pure Mathematics, by creating Research Institutes in which Fundamental Mathematics form a central part, such as the ICMAT (Instituto de Ciencias MatemΓ‘ticas, Madrid, Spain) or the IST Austria (Institute of Science and Technology Austria). Do you feel that the establishment of such a Research Institute in Portugal is crucial to promote research in Fundamental Science?Of course! It would be wonderful if Portugal had a Research Institute in Mathematics like it has in other fields, for instance the Champalimaud Centre for the Unknown or the Gulbenkian Institute of Science.
Research today in Mathematics is being influenced and shifted by the rise of neighbouring disciplines, such as Data Science, Machine Learning, Quantum Computing, Mathematical Biology, etc, and there is a big pressure to give most of the funds to Applicable Mathematics. Do you think Pure Mathematics can continue itβs path as before or, in order to strive and be funded, has to stay in close daily contact with Applied Sciences?I think it is important for Mathematics to be in contact with Applied Sciences for two reasons. One, of course, is the funding issue. The other is more philosophical and is related with the fact that if we work with physical quantities that are governed by fundamental principles, then the mathematical research arising from that tends to reach several fields of Mathematics and Applied Sciences.
For instance, the fact that minimal surfaces are physical objects (i.e., they can be observed) is one of the reasons that minimal surfaces are found across several fields of Mathematics (Geometry, Relativity, Algebraic Geometry, Dynamical Systems, etc). For instance, I ask questions and talk to colleagues of mine working in Materials Science to have a better idea of how minimal surfaces should distribute themselves in space (they call them gyroids).
What excites you most in mathematics research, and what makes you pursue problem after problem, even after solving already many famous ones? Do you want to tell us about your future projects?I have always been attracted by simple questions that an undergraduate can understand, but that in order to be solved one needs sophisticated mathematics. As for the problems that I pursue, it is a mix. Some of them I am motivated because the problems are spin-offs from a larger question that I cannot answer. Other times, I hear about some theorem or conjecture which I find fascinating and so I try to see if I can explain that to myself using the tools that I know. Most of the times I donβt, and that means I now have a new direction of research.
You often mention your interest in following portuguese mathematics. Do you see yourself returning to Portugal and establishing a new research group here?Of course. I would love to have the opportunity to establish a research group in Portugal and help the young undergraduates pursue a career in Mathematics, in the same way that people helped me when I was an undergraduate.
40
![Page 41: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/41.jpg)
configurAtion sPAces of Points
by Ricardo Campos*
* IMAG, Univ. Montpellier, CNRS, Montpellier, France. [email protected]
Given a topological space, one can consider its configuration space of n pairwise distinct points. We study the topological prop-erties of such configuration spaces and address question of homotopy invariance.
1 CONFIGURATION SPACES ANDDEFINITIONS
Let ππ be a topological space and ππ π π be an integer.The configuration space of ππ (non-overlapping) points onππ is the set
Confππ(πππ π π(ππ1, β¦ , πππππ β ππππ | ππππ β ππππ if ππ β ππππ
1β’
2β’β’3 β Conf3(Torusπ
Notice that being a subset of ππππ, the configurationspace Confππ(πππ is itself a topological space. ConfiguΒration spaces describe the state of an entire system asa single point in a higher dimensional space.
It is clear that many situations can be expressed interms of configuration spaces. For instance, in meΒchanics, where objects can often be assumed to bepoints and are not allowed to take the same place, theconfiguration of the system is a point on the configuΒration space.
What might be less clear is why we should be inΒterested in Confππ(πππ not just as a set, but also as atopological space, and in its homotopical properties.
Here are some examples where the topology ofconfiguration spaces appear:
β’ Imagine you have ππ small robots on a plane
with obstacles. The surface of movement ππ cantypically be represented as the complement ofthe obstacles. If we approximate the robots bypoints, their movement corresponds to a pathin Confππ(πππ. Turning the problem around,we can consider the path space on Confππ(πππ,Map([π, 1], Confππ(ππππ with the two projections
πππ Map([π, 1], Confππ(ππππ ππ Confππ(πππ π Confππ(πππ
given by the initial and final configuration. Amotion planning algorithm is essentially a secΒtion of the map ππ. Unless the configurationspace is contractible (which is almost never thecase) such a section does not exist globally. Thetopological complexity (surveyed in last yearsBulletin [11]) is a homotopy invariant that alΒlows us to construct notΒveryΒdiscontinuous secΒtions.
β’ In knot theory one wishes to classify all knotsup to ambient isotopy, which corresponds to theconnected components of the space of smoothembeddings Emb(ππ1, β3π. As a first approxiΒmation of the knot one can discretise the knotinto many points, which gives a particular kindof configuration on β3. In fact, a drastic genΒeralisation of this problem is the goal of unΒderstanding the homotopy type of the embedΒding space Emb(ππ, πππ between two smooth
1
CIM Bulletin December 2020.42 41
![Page 42: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/42.jpg)
manifolds. Notice that such an embeddingππ π ππ π ππ induces amap at the level of configΒuration spaces ππππ π Confππ(πππ π Confππ(πππ byevaluation pointwise. Under good conditions,the GoodwillieΒWeiss embedding calculus [2]tells us that we can recover (up to homotopy)the embedding space Emb(πππ πππ from the dataof all the ππππ together with some additional algeΒbraic structure.
β’ The pure braid group on ππ strands, denoted PBππis the group whose elements are ππ braids (up toambient isotopy), and whose group operation iscomposition of braids.
The pure braid group PBππ is isomorphic to thefundamental group of the configuration space ofππ points on the plane ππ1(Confππ(β2ππ.
β’ In quantum field theory, namely in ChernΒSiΒmons theory, one can construct invariants offramed smoothmanifolds via integrals over conΒfiguration spaces [3].
We point out that some authors would call this thespace of ordered configurations. The unordered config-uration space (or configurations of indistinguishablepoints) can be seen as the quotient space of Confππ(πππby the action of the symmetric groupππππ which acts bypermuting the π₯π₯ππβs.
In some sense unordered configuration spacescontain less information than ordered configurationspaces: as long as we can keep track of the symmetricgroup action, we can always recover the first from thelatter.
2 EXAMPLES
Configuration spaces are very simple to define, butsurprisingly hard to understand. Compare themwithππππ which we could call the configuration space of ππ pos-sibly overlapping points on ππ. In practice, most invariΒants (such as the Euler characteristic, fundamentalgroup, cohomology over a field) of ππππ can be comΒputed from the same invariant on ππ.
It is very instructive to see some examples to geta feel on configuration spaces. Notice that the babycases ππ π ππ 1 give us in all generality Confπ(πππ π πand Conf1(πππ π ππ, but this is essentially all we cansay for an arbitrary topological space.
1. For the configuration of two points in theeuclidean space, there is an identification
Conf2(βπππ π βππ Γ πππππ1 Γ β>π. This followsfrom the fact the position of the two points canbe fully determined by first giving the positionof the first point, then determining the vectorthe first point makes with the second one. Onecan picture the (ππ π 1πΒdimensional sphere πππππ1
as a the angle the points make with one another.Under this identification, the ππ2 action maps apoint in πππππ1 to its antipode. Notice that in parΒticular the unordered configuration space willbe nonΒorientable for odd ππ.
2. If we consider the graph given by connectingthree vertices to a fourth vertex, its configuraΒtion space Conf2( π is the following space
Notice that in dimension 1 there is a pheΒnomenon of nonΒlocality, in which even if twoof the points are close, it might be difficult forthem to exchange positions.
3. On the interval πΌπΌ π (ππ 1π, the configurationspace of ππ points has πππ connected components,corresponding to all possible ways to order ππpoints. All of these components are homeomorΒphic to the open ππΒsimplex
{(π₯π₯1π β¦ π π₯π₯πππ | π < π₯π₯1 < β― < π₯π₯ππ < 1}.
3 HOMOTOPY TYPE
This last example 3 is the simplest example of a topoΒlogical invariant (connected components) on the conΒfiguration space that cannot be deduced just fromthe invariant on the base space. While one could betempted to dismiss it as trivial, since it can only hapΒpen in dimension 1, it is actually a shadow of a moregeneral problem.
Indeed, these issues are related with the nonΒfunctoriality of
Confππ π Top. Spaces βΆ Top. Spaces.
242
![Page 43: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/43.jpg)
If a map ππ π ππ π ππ is not injective, the induced mapππ ππ π ππππ π ππ ππ will not restrict to the respective conΒfiguration spaces.
Recall that two spaces ππ and ππ are said to be ho-motopy equivalent, denoted ππ π ππ , if there are mapsππ π ππ π ππ and ππ π ππ π ππ such that ππ π ππ is homoΒtopic to the identity idππ in the sense that there existsa map βπ ππ π πππ ππ π ππ such that β(π¦π¦π ππ¦ π¦ π¦π¦ andβ(π¦π¦π ππ¦ π¦ ππ π ππ(π¦π¦π¦, and similarly for ππ π ππ . When wetalk about the homotopy type of ππ, we mean the equivΒalence class of all spaces homotopy equivalent to ππ.
Typically, invariants we study (and all those menΒtioned in this survey) of topological spaces dependonly on the homotopy type. The natural question thatone is led to ask is whether the homotopy type is preΒserved by taking configuration spaces, i.e., whetherππ π ππ will guarantee that Confππ(πππ¦ π Confππ(ππ π¦.This sounds very implausible given the preceding disΒcussion. Also, out of a homotopy equivalence beΒtweenππ and ππ , there seems to be no way to constructa single map relating their configuration spaces.
In fact, the very first example 1 provides already aplethora of counterΒexamples, since regardless of thedimension, all Euclidean spaces are contractible (andhence homotopy equivalent), but Conf2(βπππ¦ π ππππππ
and no two different dimensional spheres have thesame homology, cohomology or homotopy groups.
Given this, one might be surprised that the nextopen question is believed to be true.
CONJECTURE 1.β For simply connected compactmanifolds without boundary, the homotopy type ofConfππ(πππ¦ only depends on the homotopy type of ππ .
The more general conjecture for nonΒsimply conΒnected spaces was only disproved in 2005, when LonΒgoni and Salvatore were able to show that the lensspaces πΏπΏ7π2 and πΏπΏ7ππ provide a counterΒexample bycomputing the Massey products on the universal covΒers of the respective configuration spaces of 2 points[8].
In the last section I will try to provide some eviΒdence for this conjecture in the form of Theorem 7.From now on, we will restrict our study to the case ofsmooth manifolds.
4 COMPACT VERSION OF CONFIGURATIONSPACES
Suppose that ππ is a compact smooth manifold. Evenif ππ is compact, the configuration space Confππ(πππ¦
is not compact when ππ π 2, since a sequence of twopoints moving into the same place does not converge.This is an unfortunate property to lose: For instance,in situations where one wishes to consider integralsover configuration spaces (as in quantum field theΒory), one has to deal with issues of convergence.
A neat way to address this issue is to work inΒstead with a suitable compactification of Confππ(πππ¦.The most natural one is perhaps to consider ππππ, butthis could of course lose all homotopy informationas in the case of βππ. The strategy is instead to emΒbed Confππ(πππ¦ π ππ in some compact manifold withboundary ππ , such that Confππ(πππ¦ sits in ππ as its inΒterior, since manifolds with boundary are homotopyequivalent to their interiors.
The construction of such manifold is due to AxΒelrod and Singer but is usually called the Fulton-MacPherson compactification of Confππ(πππ¦, as it isa real analog of an iterated blowΒup constructionfrom algebraic geometry. This manifold is denotedFMππ(πππ¦ and admits a very visual description. InΒtuitively, instead of allowing two points moving toΒwards each other tomeet, we allow them to be infinites-imally close together, but still retaining the informationof the direction in which they collided (i.e. they canstill move around each other along a sphere of dimenΒsion dim ππ π π).
β’ β’π 2
β’3 β FM3(Torusπ¦
In the case where there are only two points, indeedFM2(πππ¦ will be a manifold with boundary, but othΒerwise one needs to distinguish different situationswhen more than two points collide, so in generalFMππ(πππ¦ will be a compact smooth manifold with cornerswhose interior is Confππ(πππ¦.
In the figure above, fixing points π and 2, thereare three possible cases when moving point 3 closeto π and 2: (A) Point 3 stays at the same βinfinitesiΒmality stratumβ; (B) Point 3 furthermore approachesinfinitesimally point π, even from the perspective ofpoint 2; (C) Similar but point 3 approaches point 2.
β’π3β’
3β’πβ’
2β’β’2(π΄π΄π¦ (πΆπΆπ¦
For details and a nice exposition see [9].
3CIM Bulletin December 2020.42 43
![Page 44: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/44.jpg)
5 THE COHOMOLOGY OF Confππ(βππ)
So far we have considered configuration spaces of afixed number of points ππ, but given the same basemanifold ππ , there are obvious relations between conΒfigurations of different number of points. Namely,given 1 β€ ππ π ππ β€ ππ, there are projectionmaps
ππππππ βΆ Confππ(ππ) π Conf2(ππ)π (1)
given by forgetting the position of all points but theππth and ππth one.
These projection maps provide us with an attemptto inductively try to understand configurations of alarge number of points from a smaller one. To illusΒtrate this idea, let us consider in more detail the caseof configurations of points in βππ, where we can get afull description of the cohomology ring of Confππ(βππ).For concreteness, let us denote by π»π»β’(ππ) the coΒhomology ring of ππ with real coefficients. SinceConfππ(ππ) is a smooth manifold, we will interpretthis graded commutative βΒalgebra as the cohomolΒogy of de Rham algebra of differential forms, denotedΞ©β’(Confππ(ππ)).
From our previous example 1 we deduce thatπ»π»β’(Conf2(βππ)) = π»π»β’(πππππ1) = β1ββππ, where ππ isa degree ππ π 1 element representing the cohomologyclass of the volume form on πππππ1.
THEOREM 2 (ARNOLD [1] AND COHEN [5]).β Thecohomology βΒalgebra π»π»β’(Confππ(βππ)) is given by
π΄π΄πππππ =Sym(ππππππ)1β€πππππβ€ππ
(ππππππ = (π1)πππππππππ ππ2ππππ = 0πArnold)
(2)
where ππππππ are elements of degree ππ π 1, Sym deΒnotes the symmetric algebra and the Arnold relationis ππππππππππππ + ππππππππππππ + ππππππππππππ = 0.
Heuristically, ππππππ represents the interaction beΒtween the points ππ and ππ, while the Arnold relaΒtions represent threeΒpoint interactions. The theoremstates that these relations generate all existing relaΒtions.
PROOF (SKETCH).β The first step is to construct themap π΄π΄πππππ β π»π»β’(Confππ(βππ)). We interpret ππππππ asan element of π»π»β’(Confππ(βππ)) by pulling back alongthe projection ππππππ from equation (1) into the (ππ π 1)Βsphere: ππππππ β ππβ
ππππππ.The relation ππ2
ππππ = 0 holds since it holds for ππ πΞ©πππ1(πππππ1). Switching the indices inππππππ correspondsto applying the antipodal map in πππππ1, which has adegree opposite to the dimension of the sphere, fromwhere it follows that ππππππ = (π1)ππππππππ.
There are various short proofs of the Arnoldrelation, none of them completely immediate.It can be deduced by analysing the fibrationππ12 βΆ Conf3(βππ) β Conf2(βππ). Alternatively, onecan explicitly find a form ππ such that ππππ = Arnoldconstructed as a fiber integral ππ = β«4 ππ14ππ24ππ34, aswe will see in the final section.
Now that we have established maps
π΄π΄πππππ β π»π»β’(Confππ(βππ))πwe need to show that they are isomorphisms, whichcan be done by induction on ππ. For this, we observethat the map Confππ(βππ) β Confπππ1(βππ) forgettingthe last point is a fibration whose fiber is homotopyequivalent to a wedge sum of spheres βπππ1
ππ=1 πππππ1 andthen apply the Serre spectral sequence. β
6 RATIONAL HOMOTOPY THEORY
Let us step back from configuration spaces for a moΒment to consider the general problem of understandΒing the homotopy type of spaces via some algebraicinvariant.
The main issue is that invariants such as the cohoΒmology ring of a space do not capture the homotopytype sufficiently faithfully. A potentially strongerinvariant is given by the higher homotopy groupsππππ, since Whitehead theorem guarantees that a mapof CWΒcomplexes ππ β ππ inducing isomorphismsππππ(ππ) β ππππ(ππ ) is a homotopy equivalence. Thisnot only does not completely solve our problem, sinceCWΒcomplexes might still have the same homotopygroups without having a map inducing an isomorΒphism, but it also has the additional issue that higherhomotopy groups are extremely difficult to computedue to their torsion parts. Rational homotopy theoryprovides a good way to address both issues, if we arewilling to work modulo torsion:
DEFINITION 3.β We say that a map of simply conΒnected spacesππ β ππ is a rational homotopy equivalenceif the induced map
ππππ(ππ) πβ€ β β ππππ(ππ ) πβ€ βis an isomorphism for all ππ. Equivalently, ππ β ππis a rational homotopy equivalence if π»π»ππ(ππ π β) βπ»π»ππ(πππ β) is an isomorphism for all ππ.
Sullivan [10] associated to a space ππ a differentialgraded (dg) commutative algebraπ΄π΄β’
ππ ππ(ππ) of piecewiselinear differential forms ππ, which the reader can thinkof as the de Rham algebra for nonΒmanifolds (and
444
![Page 45: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/45.jpg)
over β instead of β), or alternatively as the singuΒlar βΒcochains on ππ, πΆπΆβ’(πππ β) (except that the cupproduct is not commutative before passing to cohoΒmology). The cohomology of the dg algebra π΄π΄β’
ππ ππ(ππ)is the graded algebra π»π»β’(πππ β).
In the category of rational dg commutative alΒgebras, one considers the corresponding notion ofhomotopy equivalence, which is the one of a quasi-isomorphism. These are morphisms of dg commutaΒtive algebras π΄π΄β’ β π΅π΅β’ such that the induced map incohomology π»π»β’(π΄π΄) β π»π»β’(π΅π΅) is an isomorphism.
The main result of Sullivan is that this construcΒtion captures faithfully the rational homotopy type ofspaces.
CAVEAT 4.β Unlike ordinary homotopy equivaΒlences, having a rational homotopy equivalence ππ βππ does not imply the existence of a rational homoΒtopy equivalence ππ β ππ. We say that ππ and ππ arerational homotopy equivalent and we write ππ πβ ππif there is a zigΒzag of rational homotopy equivalences
ππ ππππ1 πβ β― ππππππ πβππSimilarly, quasiΒisomorphisms of dg commutative alΒgebras are not invertible so the same remark holds.
There is a general construction in category theory:Given a category ππ with some set of homotopy equiva-lences π»π» π» Morphisms(ππ ), one can construct the hoΒmotopy category of ππ , denoted ππ ππ»π»β1], which posΒsesses the same objects as ππ , but where we formallyinvert the maps in π»π» , such that they become isomorΒphisms in ππ ππ»π»β1].
THEOREM 5 ([10]).β The construction π΄π΄ππ ππ estabΒlishes an equivalence of categories
π΄π΄ππ ππ βΆ scSpacesπr.h.e.β1] β β β DGCA>1πq.i.β1]from the category of simply connected topologicalspaces of finite type up to rational homotopy equivΒalence, to the category of dg commutative βΒalgebrasof finite type concentrated in degrees > 1 up to quasiΒisomorphism.
In practice, this result allows us to study topologycompletely via (differential graded) algebraic methΒods. Any dg commutative algebra quasiΒisomorphicto π΄π΄ππ ππ(ππ) is therefore called a rational model of ππ.A classical question in rational homotopy theory iswhether one can find a small model for ππ. The smallΒest possible candidate to be a model of ππ would be itscohomology βΒalgebra, but in general it is not truethat π»π»β’(πππ β) πβ π΄π΄ππ ππ(ππ). If that happens to be the
case, we say that ππ is formal.
It should be pointed out that in the context of SulliΒvanβs theorem there is nothing special about β exceptthat it is a field of characteristic 0. Replacing it withβ we would talk about the real homotopy type of ππ inΒstead.
7 MODELS FOR CONFIGURATION SPACES
In this final section we will see how one can constructa nice model of the real homotopy type of configuraΒtion spaces using graphs. This will in particular allowus to prove the real version of Conjecture 1, see TheoΒrem 7.
THEOREM 6 ([4]).β Let ππ be a compact smoothmanifold without boundary and ππ π β. There exΒists a nice dg commutative βΒalgebra spanned by acertain type of graphs Graphsππ(ππ) modeling the realhomotopy type of Confππ(ππ). This is expressed by adirect quasiΒisomorphism of algebras into the algebraof semiΒalgebraic forms1 of FMππ(ππ):
Graphsππ(ππ) π π(FMππ(ππ))π (4)
Even though βππ is not a compact manifold, it is stillinstructive to go back to Theorem 2 and start by unΒderstanding how one could try to obtain a model ofConfππ (βππ) out of the computation of its cohomology.
The only reasonably natural attempt of estabΒlishing a quasiΒisomorphism π»π»β’(Confππ (βππ)) intoπ(FMππ(βππ)) would involve mapping ππππππ π π΄π΄πππππ intothe volume form of the spheres ππππβ1. However, sincethe Arnold relations are not satisfied at the level of difΒferential forms this cannot produce a map compatiblewith the product.
While such a quasiΒisomorphism of algebras canΒnot be directly constructed, Kontsevich [7] showedthat configuration spaces in βππ are formal by estabΒlishing a zigΒzag passing by an algebra of graphs.
Notice that one can identify the algebra π΄π΄πππππ witha graded vector space given by βΒlinear combinationsof graphs on ππ vertices, where an edge between thevertices ππ and ππ corresponds to ππππππ .
1 2 3π
β· ππ12ππ13
To be precise, depending on the parity of ππ, edgesshould be oriented or ordered, and changing orienΒtation or order (by an odd permutation) produces aminus sign, but from now on we will work up to sign.
1Pretend it is the de Rham complex. This is a minor technicality due to the lack of smoothness of forgetting points near the corners.
5CIM Bulletin December 2020.42 45
![Page 46: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/46.jpg)
Under this identification, edges have degree ππππ andthe commutative product on graphs is given by superΒposition of vertices and taking the union of edges.
The idea now is to attempt to resolve π΄π΄πππππ byadding a new kind of vertices, thatwouldmimic phanΒtom points moving freely in the configuration space.Concretely, one can define a dg commutative algebraGraphsππ(βππ), spanned by graphs with ππ labeled verΒtices as before and a arbitrary number of unlabeledvertices which now are of degree πππ.
The differential of a graph Ξ β Graphsππ(βππ) isgiven by summing all ways of contracting an unlaΒbeled vertex in Ξ along an edge. Here is an exampleexhibiting the Arnold relation as a coboundary:
πππ
2
3=
π
2
3+
π
2
3+
π
2
3β Graphs3(βππ).
Notice that the differential kills an unlabeled verΒtex and an edge so it is indeed of degree+π and (beingcareful with signs) it squares to zero. The product isstill given by superposition of labeled vertices (in parΒticular it adds the number of unlabeled vertices).
We can now produce a map into the algebra offorms
Graphsππ(βππ) βΆ Ξ©(FMππ(βππ))given by βmapping edges ππ ππ to the volume formof the sphere ππππππ
2 and integrating out the unlabeledverticesβ. In particular, the graph above yielding theArnold relation is mapped to
ππ = β«FM4βFM3
πππ4ππ24ππ34.
The only nonΒimmediate thing that needs to bechecked is the compatibility with the differential,which follows mostly from the Stokes theorem formanifolds with corners.
In fact, to prove the formality of configurationspaces in βππ, one just needs to show that the proΒjection into graphs with no unlabeled vertices is aquasiΒisomorphism, which can be achieved by a specΒtral sequence inductive argument.
While other configuration spaces over a compactmanifoldππ will not be formal (and there is no analogof Theorem 2), stretching a bit the notion of graphsone can use similar ideas to construct the dg commuΒtative βΒalgebra Graphsππ(ππ) as follows:
As a vector space, Graphsππ(ππ) is spanned bygraphs with ππ labelled vertices, some unlabeled verΒtices and vertices can be decorated by (possibly repeatΒing) reduced cohomology classes in οΏ½ΜοΏ½π»β’(ππ).
π 2 3 4
πππ πππ
ππ2 ππ3
ππ3
β Graphs4(ππ)π
The product is still given by superposition of labeledvertices. Following heuristically Kontsevich, we wishto send such graphs to differential forms in FMππ(ππ)in a way that integrates out unlabeled vertices andsending cohomology classes in π»π»β’(ππ) to represenΒtatives in Ξ©(FMπ(ππ)). To establish the map in (4),there are three main pieces:
(i) In the case of βππ, FM2(βππ) is essentially a sphere,so edges can be sent to volume forms. To whichform in Ξ©dim ππππ(FM2(ππ)) will edges be sentto?
(ii) What kind of differential mustGraphsππ(ππ) havesuch that (4) is compatible with differentials?
(iii) How to make this map a quasiΒisomorphism?
We will not address the third point and without getΒting into details, let us just say that the first point isaddressed by mapping edges to what in mathematicalphysics is called a propagator [3].
The second point is the trickiest: As far asGraphsππ(ππ) has been described, it only depends onthe cohomology ofππ , so it has no chance of even capΒturing the real homotopy type ofππ , let alone the conΒfiguration space. All this is hidden in the differential,which splits into three pieces ππ = ππcontr.+ππPoinc.+ππππππ
.A first piece ππcontr. which contracts edges as in the βππ
case, a second piece ππPoinc. which uses the PoincarΓ© duΒality pairing on π»π»β’(ππ) to split edges into two decoΒrations
Ξ ππ . = βππππβπ»π»β’(ππ) ππππ
ππβππ
ππ .
and a third piece ππππππacting only on subgraphs conΒ
sisting of unlabeled vertices which depends on thepartition function ππππ of the universal perturbativeAKSZ topological field theory on ππ . We interpretππππ as a map from vacuum graphs (fully unlabeledgraphs in Graphs0(ππ)) into real numbers.
THEOREM 7 ([4, 6]).β Letππ be a simply connectedsmooth compact manifold without boundary. The
2Abusing the notation denoting by ππππππ both the form and its class in cohomology.
646
![Page 47: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/47.jpg)
Editor-in-ChiefAna Cristina Moreira Freitas [[email protected]]
Editorial BoardAntΓ³nio Fernandes* [[email protected]]Carlos Florentino [[email protected]]Samuel Lopes [[email protected]]
*Responsible for the production and graphic design of the bulletin
AddressCentro de MatemΓ‘tica da Universidade do PortoRua do Campo Alegre, 6874169-007 PortoPortugal
The CIM Bulletin is published once a year. Material intended for publication should be sent to one of the editors. The Bulletin is available at www.cim.pt. The CIM acknowledges the financial support of FCTβFundação para a CiΓͺncia e Tecnologia.
real homotopy type of ππ determines the real homoΒtopy type of its configuration space of points.
ππ πβ ππ π ππππππ(πππ πβ ππππππ(ππππ πππ π β.
PROOF (SKETCH).β Notice that since ππ is simplyconnected π»π»1(πππ π π and therefore decorations ofgraphs in Graphsππ(πππ have degree at least 2. FurtherΒmore we can assume that π·π· π π·π·π· ππ π· π· by thePoincarΓ© conjecture.
All pieces of the defining data of Graphsππ(πππ deΒpend only on the real homotopy type of ππ , with thepossible exception of ππππππ
.One can show that ππππππ
depends only on the valueof ππππ on graphs consisting only of degree π unlaΒbeled vertices with valence π· 3 (decorations count asvalence).
The proof of the theorem now follows from thefollowing purely combinatorial statement: Using thatdecorations have degree at least 2, vertices have degreeβπ·π· and edges have degree π·π· β 1, the only π· 3Βvalentgraphs of degree π are trees.
It turns out that the values of ππππ on trees deΒpend only on the real homotopy type of ππ . It folΒlows that Graphsππ(πππ and therefore ππππππ(πππ alsodepends only on the real homotopy type of ππ . β
REFERENCES
[1] V. I. Arnold. The cohomology ring of the groupof dyed braids. Mat. Zametki, 5:227β231, 1969.
[2] P. Boavida de Brito and M. Weiss. Spacesof smooth embeddings and configuration cateΒgories. Journal of Topology, 11(1):65β143, 2018.
[3] R. Bott andA. S. Cattaneo. Integral invariants of3Βmanifolds. II. J. Differential Geom., 53(1):1β13,1999.
[4] R. Campos and T.Willwacher. A model for conΒfiguration spaces of points. ArXiV:1604.02043.
[5] F. R. Cohen. The homology of πππππ1Βspaces, ππ π·π. In The homology of iterated loop spaces, volume533 of Lecture Notes in Mathematics, pages 207β351. Springer, 1976.
[6] N. Idrissi. The LambrechtsβStanley model ofconfiguration spaces. Invent. Math, 216(1):1β68,2019.
[7] M. Kontsevich. Operads and motives in deΒformation quantization. Lett. Math. Phys.,48(1):35β72, 1999.
[8] R. Longoni and P. Salvatore. Configurationspaces are not homotopy invariant. Topology,44(2):375β380, 2005.
[9] D. P. Sinha. ManifoldΒtheoretic compactifiΒcations of configuration spaces. Selecta Math.(N.S.), 10(3):391β428, 2004.
[10] D. Sullivan. Infinitesimal computations intopology. Inst. Hautes Etudes Sci. Publ. Math.,47(47):269β331 (1978), 1977.
[11] L. Vandembroucq, Topological complexity ofsurfaces. CIM Bulletin 41: 16β20, 2019.
7CIM Bulletin December 2020.42 47
![Page 48: 2020 December - CIM](https://reader031.fdocuments.us/reader031/viewer/2022012510/6187d1411af2b45484184509/html5/thumbnails/48.jpg)
1st ed. 2019, XV, 396 p. 55 illus., 45 illus. in color 1st ed. 2015, XVIII, 772 p. 141 illus., 94 illus. in color 1st ed. 2015, XVI, 430 p. 120 illus., 78 illus. in color
1st ed. 2015, IX, 365 p. 37 illus., 25 illus. in color. 1st ed. 2015, XI, 465 p.
CIM Series in Mathematical SciencesSeries Editors: I. Fonseca, I.M.N. FigueiredoThe CIM Series in Mathematical Sciences is published on behalf of and in collaboration with the Centro Internacional de MatemΓ‘tica (CIM) in Portugal. Proceedings, lecture course material from summer schools and research monographs will be included in the new series.
Available atspringer.com/shop