WHATIS - American Mathematical Society βΊ journals βΊ notices βΊ 201611 βΊ rnoti-p1252.pdfTHE...
Transcript of WHATIS - American Mathematical Society βΊ journals βΊ notices βΊ 201611 βΊ rnoti-p1252.pdfTHE...
THE GRADUATE STUDENT SECTION
WHAT IS. . .
Symplectic GeometryTara S. Holm
Communicated by Cesar E. Silva
Symplecticstructures arefloppier thanholomorphicfunctions or
metrics.
In Euclidean geome-try in a vector spaceover β, lengths andangles are the funda-mental measurements,and objects are rigid.In symplectic geome-try, a two-dimensionalarea measurement isthe key ingredient, andthe complex numbersare the natural scalars.It turns out that sym-
plectic structures are much floppier than holomorphicfunctions in complex geometry or metrics in Riemanniangeometry.
The word βsymplecticβ is a calque introduced byHermann Weyl in his textbook on the classical groups.That is, it is a root-by-root translation of the wordβcomplexβ from the Latin roots
com β plexus,meaning βtogether β braided,β into the Greek roots withthe same meaning,
πππ β ππππ πππ Γ³π.Weyl suggested this word to describe the Lie group thatpreserves a nondegenerate skew-symmetric bilinear form.Prior to this, that Lie group was called the βline complexgroupβ or the βAbelian linear groupβ (after Abel, whostudied the group).
A differential 2-formπ on (real) manifoldπ is a gadgetthat at any point π β π eats two tangent vectors andspits out a real number in a skew-symmetric, bilinearway. That is, it gives a family of skew-symmetric bilinearfunctions
ππ βΆ πππΓπππ β β
Tara S. Holm is professor of mathematics at Cornell Universityand Notices consultant. She is grateful for the support of the Si-mons Foundation.Her e-mail address is [email protected].
For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1450
depending smoothly on the point π β π. A 2-formπ β Ξ©2(π) is symplectic if it is both closed (its exteriorderivative satisfies ππ = 0) and nondegenerate (eachfunction ππ is nondegenerate). Nondegeneracy is equiva-lent to the statement that for each nonzero tangent vectorπ£ β πππ, there is a symplectic buddy: a vector π€ β πππso that ππ(π£,π€) = 1. A symplectic manifold is a (real)manifold π equipped with a symplectic form π.
Nondegeneracy has important consequences. Purely interms of linear algebra, at any point π β πwemay choosea basis of πππ that is compatible with ππ, using a skew-symmetric analogue of the Gram-Schmidt procedure. Westart by choosing any nonzero vectorπ£1 and then finding asymplectic buddyπ€1. These must be linearly independentby skew-symmetry. We then peel off the two-dimensionalsubspace that π£1 and π€1 span and continue recursively,eventually arriving at a basis
π£1,π€1,β¦ ,π£π,π€π,which contains an even number of basis vectors. Sosymplectic manifolds are even-dimensional. This alsoallows us to think of each tangent space as a complexvector space where each π£π and π€π span a complexcoordinate subspace. Moreover, the top wedge power,ππ β Ξ©2π(π), is nowhere-vanishing, since at each tangentspace,
ππ(π£1,β¦ ,π€π) β 0.In other words,ππ is a volume form, andπ is necessarilyorientable.
Symplectic geometry enjoys connections to algebraiccombinatorics, algebraic geometry, dynamics, mathemati-cal physics, and representation theory. The key examplesunderlying these connections include:(1) π = π2 = βπ1 with ππ(π£,π€) = signed area of the
parallelogram spanned by π£ and π€;(2) π any Riemann surface as in Figure 1 with the area
for π as in (1);
β―Figure 1. The area form on a Riemannian surfacedefines its symplectic geometry.
1252 Notices of the AMS Volume 63, Number 11
THE GRADUATE STUDENT SECTION(3) π = β2π with ππ π‘π = βππ₯π β§ππ¦π;(4) π = πβπ, the cotangent bundle of any manifold
π, thought of as a phase space, with π-coordinatescoming from π being positions, π-coordinates inthe cotangent directions representing momentumdirections, and π = βπππ β§πππ;
(5) π any smooth complex projective variety with πinduced from the Fubini-Study form (this includessmooth normal toric varieties, for example);
(6) π = πͺπ a coadjoint orbit of a compact connectedsemisimple Lie groupπΊ, equipped with the Kostant-Kirillov-Souriau form π. For the group πΊ = ππ(π),this class of examples includes complex projectivespace βππβ1, Grassmannians G ππ(βπ), the full flagvariety β±β(βπ), and all other partial flag varieties.
Plenty of orientable manifolds do not admit a symplecticstructure. For example, even-dimensional spheres that areat least four-dimensional are not symplectic. The reason isthat on a compact manifold, Stokesβ theorem guaranteesthat [π] β 0 β π»2(π;β). In other words, compactsymplectic manifolds must exhibit nonzero topology indegree 2 cohomology. The only sphere with this propertyis π2.
Example (3) gains particular importance because of thenineteenth century work of Jean Gaston Darboux on thestructure of differential forms. A consequence of his workisDarbouxβs Theorem. Letπ be a two-dimensional symplec-tic manifold with symplectic form π. Then for every pointπ β π, there exists a coordinate chart π about π withcoordinates π₯1,β¦ , π₯π, π¦1,β¦ ,π¦π so that on this chart,
π =πβπ=1
ππ₯π β§ππ¦π = ππ π‘π.
Tools from the1970s and 1980s set
the stage fordramatic progress
in symplecticgeometry and
topology.
This makes pre-cise the notion thatsymplectic geome-try is floppy. InRiemannian geome-try there are localinvariants such ascurvature that dis-tinguish metrics.Darbouxβs theoremsays that symplecticforms are all lo-cally identical. Whatremains, then, areglobal topologicalquestions such as, What is the cohomology ring of aparticular symplectic manifold? and more subtle sym-plectic questions such as, How large can the Darbouxcharts be for a particular symplectic manifold?
Two tools developed in the 1970s and 1980s set thestage for dramatic progress in symplectic geometry andtopology. Marsden and Weinstein, Atiyah, and Guilleminand Sternberg established properties of the momentummap, resolving questions of the first type. Gromov intro-duced pseudoholomorphic curves to probe questions of
the second type. Let us briefly examine results of eachkind.
If a symplectic manifold exhibits a certain flavor ofsymmetries in the form of a Lie group action, then itadmits a momentum map. This gives rise to conservedquantities such as angular momentum, whence the name.The first example of a momentum map is the heightfunction on a 2-sphere (Figure 2).
Figure 2. The momentum map for π1 acting on π2 byrotation.
In this case, the conserved quantity is angular momen-tum, and the height function is the simplest example ofa perfect Morse function on π2. When the Lie group is aproduct of circles π = π1 Γβ―Γπ1, we say that the mani-fold is a Hamiltonian π-space, and the momentum mapis denoted Ξ¦ βΆ π β βπ. In 1982 Atiyah and independentlyGuillemin and Sternberg proved the Convexity Theorem(see Figure 3):Convexity Theorem. If π is a compact Hamiltonian π-space, then Ξ¦(π) is a convex polytope. It is the convex hullof the images Ξ¦(ππ) of the π-fixed points.
Figure 3. Atiyah and Guillemin-Sternberg proved thatif a symplectic manifold has certain symmetries, theimage of its momentum map is a convex polytope.
This provides a deep connection between symplecticand algebraic geometry on the one hand and discretegeometry and combinatorics on the other. Atiyahβs proofdemonstrates that the momentum map provides Morsefunctions on π (in the sense of Bott), bringing the powerof differential topology to bear on global topologicalquestions about π. Momentum maps are also used toconstruct symplectic quotients. Lisa Jeffrey will discussthe cohomology of symplectic quotients in her 2017Noether Lecture at the Joint Mathematics Meetings, andmanymore researchers will delve into these topics duringthe special session Jeffrey and I are organizing.
Through example (2), we see that two-dimensional sym-plectic geometry boils down to area-preserving geometry.Because symplectic forms induce volume forms, a naturalquestion in higher dimensions is whether symplectic ge-ometry is as floppy as volume-preserving geometry: can amanifold be stretched and squeezed in any which way so
December 2016 Notices of the AMS 1253
THE GRADUATE STUDENT SECTION
Gromov provedthat symplecticmaps are more
rigid thanvolume-
preservingones.
long as volume is pre-served? Using pseudoholo-morphic curves, Gromovdismissed this possibility,proving that symplecticmaps are more rigid thanvolume-preserving ones. Inβ2π we let π΅2π(π) denotethe ball of radius π. In1985 Gromov proved thenonsqueezing theorem (seeFigure 4):
Nonsqueezing Theorem.There is an embedding
π΅2π(π ) βͺ π΅2(π) Γ β2πβ2
preserving ππ π‘π if and only if π β€ π.One direction is straightforward: if π β€ π, then
π΅2π(π ) β π΅2(π) Γ β2πβ2. To find an obstruction tothe existence of such a map, Gromov used a pseudo-holomorphic curve in π΅2(π) Γ β2πβ2 and the symplecticembedding to produce a minimal surface in π΅2π(π ),forcing π β€ π.
On the other hand, a volume-preserving map existsfor any π and π . Colloquially, you cannot squeeze asymplectic camel through the eye of a needle.
Gromovβs work has led to many rich theories ofsymplectic invariants with pseudoholomorphic curves
Figure 4. The Nonsqueezing Theorem gives geomet-ric meaning to the aphorism on the impossibility ofpassing a camel through the eye of a needle: A sym-plectic manifold cannot fit inside a space with a nar-row two-dimensional obstruction, no matter how bigthe target is in other dimensions.This cartoon originally appeared in the article βTheSymplectic Camelβ by Ian Stewartβpublished in theSeptember 1987 issue of Natureβwe thank him forhis permission to use it. Cosgrove is a well-knowncartoonist loosely affiliated with mathematicians. Hedid drawings for Manifold for a few years and mostrecently did a few sketches for the curious cookbookSimple Scoff:https://www2.warwick.ac.uk/newsandevents/pressreleases/50th_anniversary_cookery.
the common underlying tool. The constructions rely onsubtle arguments in complex analysis and Fredholmtheory. These techniques are essential to current workin symplectic topology and mirror symmetry, and theyprovide an important alternativeperspective on invariantsof four-dimensional manifolds.
Further details on momentum maps may be found in[CdS], and [McD-S] gives anaccountofpseudoholomorphiccurves.
References[CdS] A. Cannas da Silva, Lectures on Symplectic Geometry,
Lecture Notes in Mathematics, 1764, Springer-Verlag,Berlin, 2001. MR 1853077
[McD-S] D. McDuff and D. Salamon, Introduction to Symplec-tic Topology, Oxford Mathematical Monographs, OxfordUniversity Press, New York, 1995. MR 1373431
ABOUT THE AUTHOR
In addition to mathematics, Tara Holm enjoys gardening, cooking with her family, and exploring the Finger Lakes. This column was based in part on her AMSβMAA Invited Address at MAA MathFest 2016 held in August.
Tara S. Holm
ABOUT THE AUTHOR
In addition to mathematics, Tara Holm enjoys gardening, cooking with her family, and exploring the Finger Lakes. This column was based in part on her AMSβMAA Invited Ad-dress at MathFest 2016 held in August. Tara S. Holm
Ph
oto
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A Decade Ago in the Notices: PseudoholomorphicCurves
βWHAT ISβ¦Symplectic Geometry?β discusses Gromovβsβnonsqueezing theorem,β a key result in symplecticgeometry. Gromovproved the theoremusing thenotionof a pseudoholomorphic curve, which he introduced in1986.
Readers interested in these topics might also wishto read βWHAT ISβ¦a Pseudoholomorphic Curve?β bySimon Donaldson, which appeared in the October 2005issue of the Notices. βThe notion [of pseudoholomor-phic curve] has transformed the field of symplectictopology and has a bearing on many other areas suchas algebraic geometry, string theory and 4-manifoldtheory,β Donaldson writes. Starting with the basic no-tion of a plane curve, he gives a highly accessibleexplanation of what pseudoholomorphic curves are.He notes that they have been used as a tool in sym-plectic topology in two main ways: βFirst, as geometricprobes to explore symplectic manifolds: for examplein Gromovβs result, later extended by Taubes, on theuniqueness of the symplectic structure on the complexprojective planeβ¦Second, as the source of numericalinvariants: GromovβWitten invariants.β
Another related piece βWHAT ISβ¦a Toric Variety?βby Ezra Miller appeared in the May 2008 issue ofthe Notices.
1254 Notices of the AMS Volume 63, Number 11