Overview

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First Lecture

August 30, 2019 4:00 pm
UQAM, Pavillon Sherbrooke, 200, Sherbrooke St. West, room SH-3620

"Khovanov homology, 3-manifolds, and 4-manifolds" (colloquium)

Khovanov homology is an invariant of knots in R^3. A major open problem is to extend its definition to knots in other three-manifolds, and to understand its relation to surfaces in 4-manifolds. I will discuss some partial progress in these directions, from different perspectives (gauge theory, representation theory, sheaf theory). In the process I will also review some of the topological applications of Khovanov homology.

Second Lecture

September 3, 2019
UQAM, room number coming soon

"Khovanov homology and surfaces in 4-manifolds"

Back in 2004, Rasmussen extracted a numerical invariant from Khovanov-Lee homology, and used it to give a new proof of Milnor’s conjecture about the slice genus of torus knots. In this talk, I will describe a generalization of Rasmussen’s invariant to null-homologous knots in S^1 x S^2, and prove inequalities that relate it to the genus of surfaces in D^2 x S^2 and in the complex projective space. This is based on joint work with Marco Marengon, Sucharit Sarkar, and Mike Willis.

Third Lecture

September 4, 2019
UQAM, room number coming soon

"SL(2,C) Floer homology for knots and 3-manifolds"

I will explain the construction of some new homology theories for knots and three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. These invariants are models for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres and small knots in S^3, and discuss the connection to the Kapustin-Witten equations, Khovanov homology, skein modules, and the A-polynomial. The three-manifold construction is joint work with Mohammed Abouzaid, and the one for knots is joint with Laurent Cote.

Forth Lecture

September 5, 2019
UQAM, room number coming soon

"GPV invariants and knot complements"

Gukov, Putrov and Vafa predicted (from physics) the existence of some 3-manifold invariants that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. Further, they should admit a categorification, in the spirit of Khovanov homology. Although a mathematical definition of the GPV invariants is lacking, they can be computed in many cases. In this talk I will discuss what is known about the GPV invariants, and their behavior with respect to Dehn surgery. The surgery formula involves associating to a knot a two-variable series, obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. This is based on joint work with Sergei Gukov.