师资

Stavros Garoufalidis
讲席教授
stavros@sustech.edu.cn

Quantum Topology and Hyperbolic Geometry in Da Nang, Vietnam May 27-31, 2019

Curriculum Vitae in pdf

Research lnterests:

My research interests are in low (i.e. 3 and 4) dimensional topology, the Jones polynomial, hyperbolic geometry, mathematical physics, Chern-Simons theory, string theory, M-theory, enumerative combinatorics, enumerative algebraic geometry, number theory, quantum topology, asymptotic analysis, numerical analysis, integrable systems, motivic cohomology, K-theory, Galois theory, deformation and geometric quantization.

In my early career, I got interested in TQFT (topological quantum field theory) invariants of knotted 3-dimensional objects, such as knots, braids, srting-links or 3-manifolds.

Later on, I became interested in finite type invariants (a code name for perturbative quantum field theory invariants of knotted objects). I studied their axiomatic properties, and related the various definitions to each other. A side project was to study the various filtrations of the mapping class groups, and to explicitly construct cocycles, using finite type invariants.

More recently, I have been studying the colored Jones polynomials of a knot, and its limiting geometry and topology. The colored Jones polynomials is not a single polynomial, but a sequence of them, which is known to satisfy a linear q-difference equation. Writing the equation into an operator form, and setting q=1, conjecturally recovers the A-polynomial. The latter parametrizes out the moduli space of SL(2,C) representation of the knot complement.

Another relation between the colored Jones polynomial and SL(2,C) (ie, hyperbolic) geometry is the Volume Conjecture that relates evaluations of the colored Jones polynomial to the volume of a knot. This and related conjectures fall into the problem of proving the existence of asymptotic expansions of combinatorial invariants of knotted objects. Most recently, I am working on resurgence of formal power series of knotted objects. Resuregence is a key property which (together the nonvanishing of some Stokes constant) implies the Volume Conjecture. Resurgence is intimately related to Chern-Simons perturbation theory, and produces singularities of geometric as well as arithmetic interst. Resurgence seems to be related to the Grothendieck-Teichmuller group.

In short, my interests are in low dimensional topology, geometry and mathematical physics.

 

Collaborators(54):

NamePlaceCountry
Dror Bar-NatanUniversity of TorontoCanada
Jean BellissardGeorgia Institute of TechnologyUSA
Frank CalegariThe University of ChicagoUSA
Ovidiu CostinOhio State UniversityUSA
Zsuzsanna DancsoAustralian National University, Canberra, AustraliaAustralia
Renaud DetcherryMPIM, BonnGermany
Tudor DimofteUniversity of California, DavisUSA
Jerome DuboisUniversite Paris VIIFrance
Nathan DunfieldUniversity of Illinois Urbana-ChampainUSA
Evgeny FominykhChelyabinsk State University, ChelyabinskRussia
Jeff GeronimoGeorgia Institute of TechnologyUSA
Matthias GoernerPixar Animation StudiosUSA
Mikhal GoussarovPOMI, St. PeterburgRussia
Nathan HabeggerUniversity of NantesFrance
Andrei KapaevInternational School for Advanced Studies, TriesteItaly
Craig HodgsonUniversity of MelbourneAustralia
Neil HoffmanOklahoma State university, StillwaterUSA
Rinat KashaevUniversity of GenevaSwitzerland
Christoph KoutschanJohannes Kepler UniversityAustria
Andrew KrickerNational University of SingaporeSingapore
Piotr KucharskiUniversity of Warsaw, WarsawPoland
Alexander ItsIndiana University-Purdue UniversityUSA
Yueheng LanGeorgia Institute of TechnologyUSA
Aaron LaudaUniversity of Southern CaliforniaUSA
Thang T.Q. LeGeorgia Institute of TechnologyUSA
Christine LeeUniversity of Texas at AustinUSA
Jerome LevineBrandeis UniversityUSA
Martin LoeblCharles University, PragueCzech Republic
Marcos MarinoUniversity of GeneveSwitzerland
Thomas MattmanCalifornia State UniversityUSA
Iain MoffattUniversity of South AlabamaUSA
Hugh MortonUniversity of LiverpoolUK
Hiroaki NakamuraTokyo Metropolitan UniversityJapan
Sergey NorinMcGillCanada
Tomotada OhtsukiResearch Institute for Mathematical Sciences, KyotoJapan
Michael PolyakTel-Aviv UniversityIsrael
Ionel PopescuGeorgia Institute of TechnologyUSA
James PommersheimReed CollegeUSA
Lev RozanskyUniversity of North CarolinaUSA
J. Hyam RubinsteinUniversity of MelbourneAustralia
Henry SegermanOklahoma State UniversityUSA
Alexander ShumakovitchGeorge Washington University, Washington DCUSA
Piotr SulkowskiUniversity of Warsaw, WarsawPoland
Xinyu SunTulane UniversityUSA
Vladimir TarkaevChelyabinsk State University, ChelyabinskRussia
Peter TeichnerMax Planck Institute for mathematics, BonnGermany
Morwen ThislethwaiteUniversity of Tennessee, KnoxvilleUSA
Dylan P. ThurstonUniversity of Indiana, BloomingtonUSA
Roland van der VeenUniversity of LeidenThe Netherlands
Andrei VesninSobolev Institute of Mathematics, NovosibirskRussia
Thao VuongGeorgia Institute of TechnologyUSA
Doron ZeilbergerRutgers UniversityUSA
Don ZagierMax Planck Institute, BonnGermany
Christian ZickertUniversity of MarylandUSA

 

Ph.D. student:

NamePlaceCountry
Ian MoffattUniversity of LondonUK
Roland van der VeenUniversity of AmsterdamThe Netherlands
Thao VuongGeorgia Institute of TechnologyUSA