Foam Evaluation
Institute for Computational and Experimental Research in Mathematics (ICERM)
November 5, 2021  November 7, 2021
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Friday, November 5, 2021

9:30  9:50 am EDTWorkshop RegistrationCheck In  11th Floor Collaborative Space

9:50  10:00 am EDTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

10:00  10:45 am EDTAlgebraic versus Geometric Categorification of the Alexander polynomial11th Floor Lecture Hall
 Virtual Speaker
 Anna Beliakova, Universität Zürich
 Session Chair
 Mikhail Khovanov, Columbia University
Abstract
We construct a spectral sequence from the RobertWagner gl0homology 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, LOUISHADRIEN ROBERT, AND EMMANUEL WAGNER. 
10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTKnots and quivers, HOMFLYPT and DT11th 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 DonaldsonThomas invariants of certain symmetric quivers, which was conjectured by KucharskiReinekeStošićSułkowski. I will outline a proof of this correspondence for arborescent links via quivers associated with 4ended 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 EDTLunch/Free Time

1:30  2:15 pm EDTConstructions toward topological applications of U(1) x U(1) equivariant Khovanov homology11th 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 OzsváthStipsiczSzabó'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 EDTHomotopy types for Link homology11th 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 EDTWelcome ReceptionReception  Hemenway's Patio
Saturday, November 6, 2021

10:00  10:45 am EDTFoams, Soergel bimodules and their Hochschild homology11th Floor Lecture Hall
 Virtual Speaker
 Emmanuel Wagner, University of Paris
 Session Chair
 LouisHadrien 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 LouisHadrien Robert. 
10:55  11:15 am EDTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm EDTMotivic Springer theory11th Floor Lecture Hall
 Virtual Speaker
 Catharina Stroppel, Rheinische FriedrichWilhelmsUniversität Bonn, Hausdorff Center for Mathematics
 Session Chair
 LouisHadrien Robert, University of Luxembourg

12:00  12:15 pm EDTGroup Photo11th Floor Lecture Hall

12:15  1:45 pm EDTLunch/Free Time

1:45  2:30 pm EDTpDG structures in link homology11th Floor Lecture Hall
 Speaker
 Joshua Sussan, CUNY
 Session Chair
 Mikhail Khovanov, Columbia University
Abstract
For a prime p, the WRT invariant of a 3manifold lives in a cyclotomic ring. In order to categorify such rings, Khovanov developed the machinery of pDG algebras. Building upon work of KhovanovRozansky, we discuss a pDG structure on link homology. Using ideas of Cautis, QueffelecRoseSartori, and RobertWagner, we show that it gives rise to a categorification of the Jones polynomial at a root of unity.

2:30  3:00 pm EDTCoffee Break11th Floor Collaborative Space

3:00  3:45 pm EDTComputer Bounds for KronheimerMrowka Foam Evaluation11th 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 EDTCategorical Center of Higher GeneraLightning Talks  11th Floor Lecture Hall
 Speaker
 JinCheng Guu, Stony Brook University
 Session Chair
 Mikhail Khovanov, Columbia University
Abstract
CraneYetter model is expected to be a fullyextended 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 EDTExtended CraneYetter via SkeinsLightning 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 CraneYetter TQFT using skeins. In particular, given a 4D cobordism with corners, I define a map between skein modules based on a handle decomposition. The WittenReshetikhinTuraev TQFT naturally appears as a boundary theory to the extended CY TQFT.

4:30  4:45 pm EDTAnnular link Floer homology and gl(11)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(11). In this talk, we describe how this algebraically gives rise to a gl(11) action on annular link Floer homology, viewed as the Hochschild homology—or horizontal trace—of a tangle Floer bimodule. The gl(11) 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 7, 2021

10:00  10:45 am ESTIterated wreath products and foams, with applications to field extensions, Sylvester sums, and matrix factorizations11th 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 twodimensional TQFTs coming from matrix factorizations with those in field extensions.

10:55  11:15 am ESTCoffee Break11th Floor Collaborative Space

11:15 am  12:00 pm ESTOn sl(N) link homology with mod N coefficients11th 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 ESTLunch/Free Time

1:30  2:15 pm ESTSymplectic algebraic geometry and annular link homology11th 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 2categories 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 ESTCoffee Break11th Floor Collaborative Space

3:00  3:45 pm ESTsl(2) actions on Soergel bimodules11th Floor Lecture Hall
 Virtual Speaker
 Ben Elias, University of Oregon
 Session Chair
 Alexei Oblomkov, UMASS Amherst
Abstract
BottSamelson bimodules are bimodules over a polynomial ring, whose summands are Soergel bimodules. In type A, they are commonly used in the definition of triplygraded knot homology. This polynomial ring admits an action of the lie algebra sl(2) by derivations, leading to an action on BottSamelson bimodules, and an action on morphisms between BottSamelson bimodules. The raising operator in sl(2) agrees with the differential used when equipping these categories with pdg 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.
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