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Knot homologies, BPS states, and SUSY gauge theories

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Organized by Sergei Gukov, Mikhail Khovanov, and Piotr Sulkowski
March 16 – June 12, 2015

The aim of this program is to understand new relations between knot theory, supersymmetric field theories, and string theory. Tremendous development in knot theory in recent decades led to the formulation of polynomial knot invariants, such as the Jones polynomial and its generalizations. Intimate relations between knots and physics have been found around 20 years ago in the seminal work by Witten in which he reformulated and vastly generalized the Jones polynomial in terms of Chern-Simons quantum field theory. Moreover, following the ideas of Khovanov within the last decade it has been found that knot polynomials can be categorified, meaning that they arise as the Euler characteristics of much richer homological spaces. This perspective immediately provides a new, large family of polynomial knot invariants, more generally called (colored) superpolynomials, which arise as the Poincare characteristics of these homological spaces.

Very recently it has been realized that the above mentioned results arise naturally from multiple new physics viewpoints, which provide a fascinating interpretation of mathematical phenomena and often provide a very effective computational framework. These new physics perspectives include topics such as BPS states, topological strings, differentials in homological spaces, super-A-polynomials, 3d-3d correspondence, 3-dimensional holomorphic blocks, refined Chern-Simons theory, etc.

The aim of this program is to understand relations between all these topics, and to use the language and tools they provide to find a natural interpretation of abstract mathematical formulations of knot homologies. We also plan to develop new computational tools that allow to derive an explicit form of superpolynomials, super-A-polynomial, holomorphic blocks, etc.

The weekly seminars will take place every Tuesday and Thursday in room 313 (unless otherwise specified below).

Date and Time Title Presenters
4/7 at 11:30am – Room 313 Perturbations of Khovanov-Rozansky homology Andrew Lobb
4/9 at 11:30am – Room 313 Perturbations of Khovanov-Rozansky homology, part 2 Jake Rasmussen
4/14 at 11:30pm – Room 313 Spaces attached to knots Andrew Lobb
4/16 at 11:30am – Room 313 Knots and quantum Hall effect Alexander Abanov
4/21 at 11:30am – Room 313 Introduction to Floer Homology Part 1 Simon Donaldson
4/23 at 11:30am – Room 313 Introduction to Floer HomologyPart 2 Simon Donaldson
4/28 at 11:00am – Room 313 Introduction to Floer Homology Part 3 Simon Donaldson
4/30 at 11:30am – Room 313 Homology of quandles and Yang-Baxter operators Jozef Przytycki
5/5 at 11:30am – Room 313 From Fox 3-colorings of Knots to Homology of Yang-Baxter Operators Jozef Przytycki
5/12 at 11:30am – Room 313 HOMFLY homology of knots Marko Stosic (Instituto Superior Tecnico)
5/14 at 11:30am – Room 313 Topological recursion and quantization Motohico Mulase (University of California, Davis)
5/19 at 11:30am – Room 313 HOMFLY homology of knots, part 2 Marko Stosic (Instituto Superior Tecnico)
5/21 at 11:30am – Room 313 TBA Aliakbar Daemi (SCGP)
5/26 at 11:30am – Room 313 DAHA and Application to Torus Knots, part 1 Alexei Oblomkov (University of Massachusetts Amherst)
5/28 at 11:30am – Room 313 DAHA and Application to Torus Knots, part 2 Ross Elliot
6/9 at 11:30am – Room 313 Chern-Simons knot invariants and topological strings Pichai Ramadevi
6/11 at 11:30am – Room 313 Knots invariants for virtual knots Andrey Morozov (Institute for Theoretical and Experimental Physics)