Static Knot Energy, Hopf Charge, Static Knot Energy, Hopf Charge, and Universal Growth Lawand Universal Growth Law

(joint work with Fanghua Lin)(joint work with Fanghua Lin)

Yisong YangYisong YangPolytechnic UniversityPolytechnic University

Talk given at the Chern Institute conference “Topology and Physics” iTalk given at the Chern Institute conference “Topology and Physics” in memory of Xiaosong Linn memory of Xiaosong Lin

July 27 – 31, 2007July 27 – 31, 2007

Xiaosong’s influence outside his specialty area (loXiaosong’s influence outside his specialty area (low dimensional topology and knot invariants)w dimensional topology and knot invariants)

Got to know Xiaosong during 1993-1994 at IASGot to know Xiaosong during 1993-1994 at IAS

Initial impressions about XS:

A person of high intellectual pursuit

An inspiring educator

Genuine

A gentleman of soft voice and smile

(these impressions never changed over time)

Later encounters: conferencesLater encounters: conferences

Two most notable conferences XS Two most notable conferences XS and I had together areand I had together are

The 2nd Sun Zi Institute ConferenThe 2nd Sun Zi Institute Conference at Rutgers in 1995ce at Rutgers in 1995

The 23rd International Conference The 23rd International Conference on Differential Geometry Methoon Differential Geometry Metho

ds in Theoreticalds in Theoretical Physics at NaPhysics at Nankai in 2005nkai in 2005

Title of Rutgers Talk: Knot Energies Title of Rutgers Talk: Knot Energies and Knot Invariants (1995)and Knot Invariants (1995)

Xiaosong’s opening words: “Knots are fascinating Xiaosong’s opening words: “Knots are fascinating objects.”objects.”

Based on this talk, Xiaosong wrote a beautiful survBased on this talk, Xiaosong wrote a beautiful survey article under the same title (presenting historiey article under the same title (presenting historical developments and current state of art of the cal developments and current state of art of the subject)subject)

See: See: http://arxiv.org/PS_cache/q-alg/pdf/9604/9604018v1.pdfhttp://arxiv.org/PS_cache/q-alg/pdf/9604/9604018v1.pdf

Added impressions: A leader and an ambassador Added impressions: A leader and an ambassador of knot theoryof knot theory

Nankai Conference 2005Nankai Conference 2005

Had numerous discussions wiHad numerous discussions with Xiaosong on the knot stuth Xiaosong on the knot study Fanghua Lin and I were dy Fanghua Lin and I were doingdoing

In particular, Xiaosong told us In particular, Xiaosong told us about the work of some knabout the work of some knot theory people on “ideal kot theory people on “ideal knots” related to our work (in nots” related to our work (in particular, that of Canterellparticular, that of Canterella, Kusner, and Sullivan; Bua, Kusner, and Sullivan; Buck)ck)

Extensive email exchanges laExtensive email exchanges lasted until May 6, 2006 (oftsted until May 6, 2006 (often long and precise)en long and precise)

August, 2005

Last Email From Xiaosong Dated 05/06/2006Last Email From Xiaosong Dated 05/06/2006

Thank you very much for Thank you very much for sending me this nice sending me this nice survey. It is my hope to survey. It is my hope to getgetto know this subject to know this subject better and communicate better and communicate your work to the knot your work to the knot theorytheorycommunity.community.

Best regards,Best regards,

Xiao-SongXiao-Song

Email of Fanghua Lin on XiaosongEmail of Fanghua Lin on Xiaosong

Xiaosong and I met only a few times in person, but impressions areXiaosong and I met only a few times in person, but impressions aredeep. We shared the same last name (so God knows we may bdeep. We shared the same last name (so God knows we may be connected through a simple homotopy - that was a joke we use connected through a simple homotopy - that was a joke we used to make).ed to make).What I admire Xiaosong most is not his mathematics (I knew he What I admire Xiaosong most is not his mathematics (I knew he was one of thewas one of thebest in his business if not the best) because I did not know his stbest in his business if not the best) because I did not know his stuffuffwell. I admire his honesty and straightforwardness toward friendwell. I admire his honesty and straightforwardness toward friends ands andcolleagues. Always a very sincere, trustable and reliable friend.colleagues. Always a very sincere, trustable and reliable friend.      

At the 1997 Bekerley meeting (by NSF China, in honor of Chern), we At the 1997 Bekerley meeting (by NSF China, in honor of Chern), we decided todecided tohave a journal “Comm. in Contem. Math". I was asked to be co-have a journal “Comm. in Contem. Math". I was asked to be co-chief editor. But due to my many commitments, I turned it down.chief editor. But due to my many commitments, I turned it down. Instead I suggested Luis Caffarelli (he could not do it either). Th Instead I suggested Luis Caffarelli (he could not do it either). Then I suggested Haim Brezis.  Thinking back, I feel bad as it gaven I suggested Haim Brezis.  Thinking back, I feel bad as it gave a lot of work to Xiaosong. Hee a lot of work to Xiaosong. Hemade the journal very successful within a few years. As an editomade the journal very successful within a few years. As an editor, I didr, I didhelp him with a lot of work as well as handling some difficulthelp him with a lot of work as well as handling some difficultcases. Through these occasions, I got to know Xiaosong better. cases. Through these occasions, I got to know Xiaosong better. ToTothis day, I still miss him terribly as a colleague and a friend.this day, I still miss him terribly as a colleague and a friend.But he will always be remembered.But he will always be remembered.   Best   Best,,

FanghuaFanghua

PS. I am glad you are there for this event. Our latest work owes also tPS. I am glad you are there for this event. Our latest work owes also to Xiaoong's input.o Xiaoong's input.

Faddeev Knots: Existence of Knots Faddeev Knots: Existence of Knots from the First Principle from the First Principle

Numerical work of Battye and Sutcliffe

Faddeev’s quantum field theory model

Seminal numerical work by Faddeev and Niemi

Progress reported at Nankai in 2005 with Xiaosong present

Latest unfolding of new results

(Chalk presentation starts from here)