A ``skein relation'' can be viewed as a linear relation satisfied by the
R-matrix for a quantum group; one of the first uses of skein relations was
to give a combinatorial construction of Reshetikhin-Turaev invariants of
knots in S^3. The Hall algebra of an abelian (or triangulated) category
"counts extensions" in the category. We briefly describe how skein relations
appear in the Hall algebra of coherent sheaves of an elliptic curve, the
Hall algebra of the Fukaya category of a surface, and factorization homology
of a surface. No familiarity with the objects mentioned above will be
assumed for the talk.

The art of using quantum field theory to derive mathematical
results often lies in a mysterious transition between infinite dimensional
geometry and finite dimensional geometry. In this talk we describe a general
mathematical framework to study the quantum geometry of sigma-models when
they are effectively localized to small fluctuations around constant maps.
We illustrate how to turn the physics idea of exact semi-classical
approximation into a geometric set-up in this framework, using Gauss-Manin
connection. This leads to a theory of “counting constant maps” in a
nontrivial way.  We explain this program by a concrete example of
topological quantum mechanics and show how “counting constant loops”  leads
to a simple proof of the algebraic index theorem.

Talks at UCLA.  Please contact Li-Sheng Tseng if you plan to attend and would like to carpool.

Peter Lambert-Cole (Georgia Tech): Bridge trisections and the Thom conjecture
The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.

James Conway (UC Berkeley): Classifying contact structures on hyperbolic 3-manifolds
Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations.  Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories.  Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual.