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Organizing Committee
Columbia University
University of Southern California
University of Luxembourg
Abstract
The purpose of this workshop is to bring together mathematicians
interested in foams and their use in low-dimensional topology,
representation theory, categorification, mathematical physics, and
combinatorics. The workshop will focus on the foam evaluation formula
and its applications. More concretely, we aim to:
(a) Give a more intrinsic definition of the foam evaluation, in order, for
instance, to find similar formulas for the other Lie types;
(b) Understand the interplay between foams and matrix factorizations and
further use foams for a unified and comprehensive approach to
Khovanov-Rozansky link homology theories;
(c) Compare combinatorial foam evaluation with the geometric structures and
invariants coming from gauge theory and symplectic geometry;
(d) Study potential applications of the foamy definition of link homology
theories.
This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.
interested in foams and their use in low-dimensional topology,
representation theory, categorification, mathematical physics, and
combinatorics. The workshop will focus on the foam evaluation formula
and its applications. More concretely, we aim to:
(a) Give a more intrinsic definition of the foam evaluation, in order, for
instance, to find similar formulas for the other Lie types;
(b) Understand the interplay between foams and matrix factorizations and
further use foams for a unified and comprehensive approach to
Khovanov-Rozansky link homology theories;
(c) Compare combinatorial foam evaluation with the geometric structures and
invariants coming from gauge theory and symplectic geometry;
(d) Study potential applications of the foamy definition of link homology
theories.
This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.
This workshop is fully funded by a Simons Foundation Targeted Grant to Institutes.
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Workshop Schedule
Friday, November 05, 2021
9:30 - 9:50 AM EDT
Workshop Registration
Check In - 11th Floor Collaborative Space
9:50 - 10:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
10:00 - 10:45 AM EDT
Algebraic versus Geometric Categorification of the Alexander polynomial
11th Floor Lecture Hall
Virtual Speaker
Anna Beliakova, Universität Zürich
Session Chair
Mikhail Khovanov, Columbia University
Abstract
We construct a spectral sequence from the Robert-Wagner gl0-homology to the knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules.
This is a joint work with KRZYSZTOF K. PUTYRA, LOUIS-HADRIEN ROBERT, AND EMMANUEL WAGNER.
This is a joint work with KRZYSZTOF K. PUTYRA, LOUIS-HADRIEN ROBERT, AND EMMANUEL WAGNER.
10:55 - 11:15 AM EDT
Coffee Break
11th Floor Collaborative Space
11:15 AM - 12:00 PM EDT
Knots and quivers, HOMFLYPT and DT
11th Floor Lecture Hall
Virtual Speaker
Paul Wedrich, Universität Hamburg
Session Chair
Mikhail Khovanov, Columbia University
Abstract
I will describe a surprising connection between the colored HOMFLYPT polynomials of knots and the motivic Donaldson-Thomas invariants of certain symmetric quivers, which was conjectured by Kucharski-Reineke-Stošić-Sułkowski. I will outline a proof of this correspondence for arborescent links via quivers associated with 4-ended tangles, which is joint work with Marko Stošić. The underlying idea is to perform web evaluation simultaneously at all (exterior) colors using generating functions. It is tempting to speculate whether this idea carries over to foam evaluation.
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:15 PM EDT
Constructions toward topological applications of U(1) x U(1) equivariant Khovanov homology
11th Floor Lecture Hall
Virtual Speaker
Melissa Zhang, University of Georgia
Session Chair
Aaron Lauda, University of Southern California
Abstract
In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still unclear.
Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó's Upsilon. In this talk, I will describe the construction and properties of such an invariant. I will also discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.
This is based on joint work with Ross Akhmechet.
Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó's Upsilon. In this talk, I will describe the construction and properties of such an invariant. I will also discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.
This is based on joint work with Ross Akhmechet.
2:30 - 3:15 PM EDT
Homotopy types for Link homology
11th Floor Lecture Hall
Virtual Speaker
Nitu Kitchloo, Johns Hopkins University,
Session Chair
Aaron Lauda, University of Southern California
Abstract
I will motivate the existence of homotopy types that lift link invariants. We will briefly review recent joint work with M.Khovanov on deformations of Foam evaluations using formal group laws. This deformation suggests that (complex oriented) cohomology theories seem to be making an appearance via their evaluation on spaces (or spectra) that lift Foams. We will offer some evidence that suggests that such spectra exist.
3:00 - 4:30 PM EDT
Welcome Reception
Reception - Hemenway's Patio
Saturday, November 06, 2021
10:00 - 10:45 AM EDT
Foams, Soergel bimodules and their Hochschild homology
11th Floor Lecture Hall
Virtual Speaker
Emmanuel Wagner, University of Paris
Session Chair
Louis-Hadrien Robert, University of Luxembourg
Abstract
I will present a complete foam definition of Soergel bimodules, their morphisms and their Hochschild homology.
This is a joint work with Mikhail Khovanov and Louis-Hadrien Robert.
This is a joint work with Mikhail Khovanov and Louis-Hadrien Robert.
10:55 - 11:15 AM EDT
Coffee Break
11th Floor Collaborative Space
11:15 AM - 12:00 PM EDT
Motivic Springer theory
11th Floor Lecture Hall
Virtual Speaker
Catharina Stroppel, Rheinische Friedrich-Wilhelms-Universität Bonn, Hausdorff Center for Mathematics
Session Chair
Louis-Hadrien Robert, University of Luxembourg
12:00 - 12:15 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:15 - 1:45 PM EDT
Lunch/Free Time
1:45 - 2:30 PM EDT
p-DG structures in link homology
11th Floor Lecture Hall
Speaker
Joshua Sussan, CUNY
Session Chair
Mikhail Khovanov, Columbia University
Abstract
For a prime p, the WRT invariant of a 3-manifold lives in a cyclotomic ring. In order to categorify such rings, Khovanov developed the machinery of p-DG algebras.
Building upon work of Khovanov-Rozansky, we discuss a p-DG structure on link homology. Using ideas of Cautis, Queffelec-Rose-Sartori, and Robert-Wagner, we show that it gives rise to a categorification of the Jones polynomial at a root of unity.
2:30 - 3:00 PM EDT
Coffee Break
11th Floor Collaborative Space
3:00 - 3:45 PM EDT
Computer Bounds for Kronheimer-Mrowka Foam Evaluation
11th Floor Lecture Hall
Speaker
David Boozer, Princeton University
Session Chair
Joshua Sussan, CUNY
Abstract
Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor J^sharp, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement J^flat for J^sharp. Of particular interest is the relationship between the dimension of J^flat(K) for a web K and the number of Tait colorings Tait(K) of K; these two numbers are known to be identical for a special class of "reducible" webs, but whether this is the case for nonreducible webs is not known. We describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of J^flat(K) for a given web K, in some cases determining these quantities uniquely. We present results for a number of nonreducible example webs. For the dodecahedral web W_1 the number of Tait colorings is Tait(W_1) = 60, but our results suggest that dim J^flat(W_1) = 58.
4:00 - 4:15 PM EDT
Categorical Center of Higher Genera
Lightning Talks - 11th Floor Lecture Hall
Speaker
Jin-Cheng Guu, Stony Brook University
Session Chair
Mikhail Khovanov, Columbia University
Abstract
Crane-Yetter model is expected to be a fully-extended topological quantum field theory that categorifies the Jones polynomial. We will present its categorical values for the spaces of (co)dimension 2.
4:15 - 4:30 PM EDT
Extended Crane-Yetter via Skeins
Lightning Talks - 11th Floor Lecture Hall
Speaker
Ying Hong Tham, Albert Einstein College of Medicine
Session Chair
Mikhail Khovanov, Columbia University
Abstract
I will define an extended Crane-Yetter TQFT using skeins. In particular, given a 4D cobordism with corners, I define a map between skein modules based on a handle decomposition. The Witten-Reshetikhin-Turaev TQFT naturally appears as a boundary theory to the extended CY TQFT.
4:30 - 4:45 PM EDT
Annular link Floer homology and gl(1|1)
Lightning Talks - 11th Floor Lecture Hall
Speaker
C.-M. Michael Wong, Dartmouth College
Session Chair
Mikhail Khovanov, Columbia University
Abstract
In earlier work by Ellis, Petkova, and Vertesi, tangle Floer bimodules (a combinatorial generalization of link Floer homology) are shown to decategorify to the Reshetikhin–Turaev invariants arising in the representation theory of gl(1|1). In this talk, we describe how this algebraically gives rise to a gl(1|1) action on annular link Floer homology, viewed as the Hochschild homology—or horizontal trace—of a tangle Floer bimodule. The gl(1|1) action turns out to have an interpretation as a known basepoint action in the holomorphic Floer theory. This is based on joint work in progress with Andy Manion and Ina Petkova.
Sunday, November 07, 2021
10:00 - 10:45 AM EST
Iterated wreath products and foams, with applications to field extensions, Sylvester sums, and matrix factorizations
11th Floor Lecture Hall
Speaker
Mee Seong Im, United States Naval Academy
Session Chair
David Boozer, Princeton University
Abstract
I will explain how patched surfaces with defect circles and foams relate to separable field extensions and Galois theory, and describe a connection between overlapping foams and Sylvester double sums. I will also compare traces in two-dimensional TQFTs coming from matrix factorizations with those in field extensions.
10:55 - 11:15 AM EST
Coffee Break
11th Floor Collaborative Space
11:15 AM - 12:00 PM EST
On sl(N) link homology with mod N coefficients
11th Floor Lecture Hall
Speaker
Joshua Wang, Harvard University
Session Chair
David Boozer, Princeton University
Abstract
An interesting aspect of Khovanov homology is that it often behaves differently when coefficients are taken in a ring of characteristic 2. I'll explain a generalization of one instance of this phenomenon to sl(P) link homology in characteristic P when P is prime. The proof uses an operator defined on sl(N) link homology for any N when coefficients are taken in a ring whose characteristic divides N.
12:00 - 1:30 PM EST
Lunch/Free Time
1:30 - 2:15 PM EST
Symplectic algebraic geometry and annular link homology
11th Floor Lecture Hall
Virtual Speaker
Lev Rozansky, University of North Carolina at Chapel Hill
Session Chair
Alexei Oblomkov, UMASS Amherst
Abstract
In a joint work with A. Oblomkov we study how link homology is related to 2-categories associated with symplectic varieties: a `commuting variety’ and a Hilbert scheme of points on C^2. I will explain the basics of our construction and its relation to the annular link homology following the work of Rina Anno and Mina Aganagic.
2:30 - 3:00 PM EST
Coffee Break
11th Floor Collaborative Space
3:00 - 3:45 PM EST
sl(2) actions on Soergel bimodules
11th Floor Lecture Hall
Virtual Speaker
Ben Elias, University of Oregon
Session Chair
Alexei Oblomkov, UMASS Amherst
Abstract
Bott-Samelson bimodules are bimodules over a polynomial ring, whose summands are Soergel bimodules. In type A, they are commonly used in the definition of triply-graded knot homology. This polynomial ring admits an action of the lie algebra sl(2) by derivations, leading to an action on Bott-Samelson bimodules, and an action on morphisms between Bott-Samelson bimodules. The raising operator in sl(2) agrees with the differential used when equipping these categories with p-dg structures. A major open question is whether this leads to a consistent action of sl(2) on Soergel bimodules, as the idempotents used to project to these summands are not invariant under sl(2). If so, this has a number of interesting implications.