Organizing Committee
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.

Image for "Foam Evaluation"

Confirmed Speakers & Participants

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Friday, November 5, 2021
  • 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.
  • 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.
  • 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 6, 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.
  • 10:55 - 11:15 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:15 am - 12:00 pm EDT
    TBA
    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
    11th Floor Lecture Hall
  • 12:15 - 1:45 pm EDT
    Lunch/Free Time
  • 1:45 - 2:30 pm EDT
    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
    • Joshua Sussan, CUNY
    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.
  • 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.
Sunday, November 7, 2021
  • 10:00 - 10:45 am EST
    p-DG structures in link homology
    11th Floor Lecture Hall
    • Speaker
    • Joshua Sussan, CUNY
    • Session Chair
    • David Boozer, Princeton 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.
  • 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.

All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).

All event times are listed in .

Application Information

ICERM welcomes applications from faculty, postdocs, graduate students, industry scientists, and other researchers who wish to participate. Some funding may be available for travel and lodging. Graduate students who apply must have their advisor submit a statement of support in order to be considered.

Financial Support

Acceptable Costs
  • 1 roundtrip between your home institute and ICERM
  • Flights in economy class to either Providence airport (PVD) or Boston airport (BOS)
  • Ground Transportation to and from airports and ICERM.
Unacceptable Costs
  • Seats in economy plus, business class, or first class
  • Change ticket fees of any kind
  • Multi-use bus passes
  • Meals or incidentals
Advance Approval Required
  • Personal car travel to ICERM from outside New England
  • Multiple-destination plane ticket; does not include layovers to reach ICERM
  • Arriving or departing from ICERM more than a day before or day after the program
  • Multiple trips to ICERM
  • Rental car to/from ICERM
  • Arriving or departing from airport other than PVD/BOS or home institution's local airport
  • 2 one-way plane tickets to create a roundtrip (often purchased from Expedia, Orbitz, etc.)
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