Physics and mathematics of knot homologies: June 1 – 5, 2015

Organized by Eugene Gorsky, Sergei Gukov, Mikhail Khovanov, Vivek Shende, and Piotr Sulkowski

Dates: June 1 – 5, 2015

Attendee List Download Talk ScheduleView Videos

The aim of this workshop is to exchange current ideas in physics and mathematics related to knot homologies. Intimate relations between knots and physics, found around 25 years ago in the seminal work by Witten, grew into a fascinating web of correspondences involving topological and supersymmetric quantum field and string theories. In mathematics, (perhaps) the most important modern idea in knot theory is the program of categorification and formulation of knot homologies; this program is still being completed, and one of outstanding questions related to knot homologies is their geometric meaning.

In the last decade several knot homology theories has been developed. Some of them, such as Heegaard-Floer homology, are based on the ideas from symplectic geometry and carry a lot of deep information about the geometry of a given knot. Several other theories, developed by Khovanov and Rozansky, are more combinatorial and are defined in terms of a knot diagram. It is expected that Khovanov-Rozansky homology carry some geometric information too: for example, a slice genus bound in terms of Khovanov homology has been found. Nonetheless, while several geometric interpretations for the Khovanov-Rozansky homologies has been developed, these approaches share the same problem: the size of the chain complex grows very fast with the complexity of a knot and by far outgrow the size of its homology. As a result, no cleargeometric understanding of the homology generators is known.

Recently progress in this direction has been made for algebraic knots, where an explicit conjectural description of Khovanov-Rozansky homology in terms of the homology of Hilbert schemes of points on a singular curves has been found. This conjecture was partially verified on the level of Euler characteristics. For torus knots, further works revealed the intimate relation between these Hilbert schemes and representation theory of double affine Hecke and rational Cherednik algebras.

In parallel with all these developments, many new ideas related to knot homologies have been established in physics. Firstly, the so-called 3d-3d correspondence has been formulated, which relates Chern-Simons theory on a knot complement to a 3-dimensional supersymmetric N=2 gauge theory. Secondly, it has been found that partition functions and indices of these 3d N=2 theories are built from universal holomorphic blocks. Thirdly, a new class of algebraic curves associated to knots, so-called super-A-polynomials, akin to Seiberg-Witten curves, has been discovered. Furthermore, existence of an intricate structure of differentials in knot homologies has been postulated based on properties of BPS states, and a refinement of Chern-Simons theory, responsible for the homological grading, has been formulated. All these developments are very interesting physically, and relate to the mathematical conjectures mentioned above.

The aim of this workshop is to provide an up to date summary of all the above approaches, to formulate a list of (currently) most important problems, and to propose paths which might lead to their solution. We believe that the workshop will be a source of mutual inspirations for physicists and mathematicians, and hope that the combined effort of both our communities will lead to solutions of those problems in the foreseeable future.

This workshop is a part of the Spring 2015 program, Knot homologies, BPS states, and SUSY gauge theories, which is organized by Sergei Gukov, Mikhail Khovanov, and Piotr Sulkowski. This program takes place from March 16 – June 12, 2015.

Physics and mathematics of knot homologies Schedule


Time Title Presenters Video
9:30am Deformations of type A link homologies Paul Wedrich video
slides
10:30am Coffee Break SCGP Cafe
11:00am Knot contact homology, string topology, and the knot group Lenhard Ng video
12:00pm Lunch SCGP Cafe
2:15pm The Jones Polynomial and Khovanov Homology From Gauge Theory Edward Witten video
slides
3:30pm Coffee Break SCGP Cafe
4:00pm Experimental Results in Quiver Representation Theory Clay Cordova video

Time Title Presenters Video
9:30am Tensor structures for 2-representations Raphael Rouquier video
slides
10:30am Coffee Break SCGP Cafe
11:00am Khovanov Homology from Floer cohomology Mohammed Abouzaid video
12:00pm Lunch SCGP Cafe
1:00pm Knots and String Duality Mina Aganagic video
slides
3:30pm Coffee Break SCGP Cafe
4:00pm Knot Invariants from Topological Recursion on
Augmentation Varieties
Albrecht Klemm video

Time Title Presenters Video
9:30am Knot invariants, A-polynomials, BPS states Satoshi Nawata video
slides
10:30am Coffee Break SCGP Cafe
11:00am Hopf link and instanton calculus Amer Iqbal video
slides
12:00pm Lunch SCGP Cafe
1:15pm 2d defects in D=4, N=4 YM and triply graded link homology. Lev Rozansky video
2:30pm Khovanov homotopy types and the Burnside category Robert Lipshitz video
3:30pm Coffee Break SCGP Cafe

Time Title Presenters Video
9:30am Knot Clusters Eric Zaslow video
10:30am Coffee Break SCGP Cafe
11:00am Knot invariants arising from homological operations on Khovanov homology Alexander Shumakovitch video
12:00pm Lunch SCGP Cafe
2:15pm Quiver gauge theories and integrable lattice models Junya Yagi video
slides
3:30pm Coffee Break SCGP Cafe
4:00pm Moduli spaces, boundary conditions, and interfaces in 3d N=4 theory Tudor Dimofte video removed

Time Title Presenters Video
9:30am Hecke algebras, the torus, and knots Peter Samuelson video
slides
10:30am Coffee Break SCGP Cafe
11:00am Advances in knot polynomials Alexi Morozov video
12:00pm Lunch SCGP Cafe
1:00pm Colored HOMFLY-PT homology of knots and links, and recursion relations Marco Stosic video
2:30pm Knot homology of torus knots Alexei Oblomkov video
3:30pm Coffee Break Room 515


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