Contents
Quantum Topology and Hyperbolic Geometry in Da Nang, Vietnam
May 27-31, 2019
Curriculum
Vitae in pdf
Research
Interests
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)
Name |
Place |
Country |
Dror Bar-Natan
|
University of Toronto
|
Canada |
Jean Bellissard
|
Georgia Institute of Technology
|
USA |
Frank Calegari |
The University of Chicago |
USA |
Ovidiu
Costin |
Ohio State University |
USA |
Zsuzsanna Dancso |
Australian National University, Canberra, Australia |
Australia |
Renaud Detcherry
|
MPIM, Bonn
|
Germany |
Tudor Dimofte |
University of California, Davis |
USA |
Jerome
Dubois |
Universite Paris
VII |
France |
Nathan Dunfield |
University of Illinois
Urbana-Champain |
USA |
Evgeny Fominykh |
Chelyabinsk State University, Chelyabinsk |
Russia |
Jeff Geronimo |
Georgia Institute of
Technology |
USA |
Matthias Goerner |
Pixar Animation Studios |
USA |
Mikhal Goussarov |
POMI, St. Peterburg |
Russia |
Nathan Habegger |
University of
Nantes |
France |
Andrei Kapaev |
International School for Advanced Studies, Trieste |
Italy |
Craig Hodgson |
University of Melbourne |
Australia |
Neil Hoffman |
Oklahoma State university, Stillwater |
USA |
Rinat Kashaev |
University of Geneva |
Switzerland |
Christoph
Koutschan |
Johannes Kepler University |
Austria |
Andrew Kricker |
National University of
Singapore |
Singapore |
Piotr Kucharski |
University of Warsaw, Warsaw |
Poland |
Alexander Its |
Indiana
University-Purdue University |
USA |
Yueheng Lan |
Georgia Institute of
Technology |
USA |
Aaron Lauda |
University of Southern California |
USA |
Thang T.Q. Le |
Georgia Institute of
Technology |
USA |
Christine Lee |
University of Texas at Austin |
USA |
Jerome Levine |
Brandeis University |
USA |
Martin Loebl |
Charles University, Prague |
Czech Republic |
Marcos
Marino |
University of Geneve |
Switzerland |
Thomas
Mattman |
California State University |
USA |
Iain
Moffatt |
University of South
Alabama |
USA |
Hugh Morton |
University of Liverpool |
UK |
Hiroaki
Nakamura |
Tokyo
Metropolitan University |
Japan |
Sergey Norin |
McGill |
Canada |
Tomotada
Ohtsuki |
Research Institute
for Mathematical Sciences, Kyoto |
Japan |
Michael Polyak |
Tel-Aviv University |
Israel |
Ionel Popescu |
Georgia Institute of
Technology |
USA |
James Pommersheim |
Reed College |
USA |
Lev Rozansky |
University of North Carolina |
USA |
J. Hyam Rubinstein |
University of Melbourne |
Australia |
Henry Segerman |
Oklahoma State University |
USA |
Alexander Shumakovitch |
George Washington University,
Washington DC |
USA |
Piotr Sulkowski |
University of Warsaw, Warsaw |
Poland |
Xinyu Sun |
Tulane University |
USA |
Vladimir Tarkaev |
Chelyabinsk State University, Chelyabinsk |
Russia |
Peter
Teichner |
Max Planck Institute for
mathematics, Bonn |
Germany |
Morwen Thislethwaite |
University of Tennessee,
Knoxville |
USA |
Dylan P. Thurston |
University of Indiana, Bloomington |
USA |
Roland van der Veen |
University of Leiden |
The Netherlands |
Andrei Vesnin |
Sobolev Institute of Mathematics, Novosibirsk |
Russia |
Thao Vuong |
Georgia Institute of
Technology |
USA |
Doron
Zeilberger |
Rutgers University |
USA |
Don Zagier |
Max Planck Institute, Bonn |
Germany |
Christian
Zickert |
University of Maryland |
USA |
Ph.D.
students
Submission
to the Journal of Knot Theory and its Ramifications
- I am an associate editor for the JKTR
- You may submit your paper for JKTR here
Seminars
A
list of seminars in the MPIM and Bonn.
Miscallanea