2023 Simons Collaboration on New Structures in LowDimensional Topology Annual Meeting
Organizers:
Aaron Lauda, University of Southern California
Sergei Gukov, Caltech
Speakers:
Sergei Gukov, Caltech
Peter Kronheimer, Harvard University
Aaron Lauda, USC
Lisa Piccirillo, MIT
Raphaël Rouquier, UCLA
Mikhail Khovanov, Columbia
Peter Ozsváth, Princeton
Zoltán Szabó, Princeton
Meeting Goals:
The goal of this meeting was to build on and uncover new structures in the rapidly evolving field of lowdimensional topology. We aimed to emphasize newfound interactions with a wide array of other mathematical disciplines and mathematical physics, bringing tools from categorification and representation theory, homological algebra, symplectic geometry, contact geometry, gauge theory, and analysis. Talks in this workshop engaged researchers from across these disciplines, forging a common language and understanding of key challenges. A focus was made on highlighting future directions for collaborations and opportunities for interaction between perspectives.

Meeting Report
The first annual meeting was the true beginning of a new collaboration. A total of about hundred researchers from different areas of lowdimensional topology working on various aspects of the categorification program were brought together to establish new connections and learn about areas distant from one’s own. Excitement was definitely in the air.
The main goal of this first meeting was to forge a common language and to understand the key challenges, in order to facilitate an exchange of ideas and to explore new opportunities for collaboration. While we intend to continue to develop and evolve the collaboration, based on the lively discussions during the breaks and in the main program, the first step toward this goal was successful.
Aaron Lauda opened the scientific program with a survey of some of the main challenges that the Collaboration is going to tackle and then presented a new strategy based on reinterpreting Heegaard Floer homology as a categorified quantum invariant associated to the super Lie algebra \(gl(11)\). Since much is known about Heegaard Floer theory, its representation theoretic description can be used as a springboard for constructing more general homology theories of 3manifolds, associated with other root systems.
The topic of algebraic structures relevant to Heegaard Floer homology and its many variants continued through most of the first day, in talks from Peter Ozsvath, Zoltan Szabo, and Raphael Rouquier. In particular, Peter Ozsvath presented an ongoing project with Robert Lipshitz and Dylan Thurston, generalizing their earlier work on bordered Heegaard Floer homology from the specialized (\(U=0\)) case to the unspecialized version. These two versions are related, respectively, to the simpler (“hat”) and more advanced (“minus”) versions of Heegaard Floer homology. For the unspecialized Heegaard Floer homology, Peter Ozsvath gave an explicit combinatorial description of a weighted \(A_{\infty}\) algebra associated to a 2torus and described higher products and relations in this algebra. He then explained how one can construct modules associated to 3manifolds with torus boundary and outlined various directions for future work.
Zoltan Szabo’s talk was devoted to similar versions of knot Floer homology and their bordered algebras: \(B\), \(C\), and \(P\). The latter, called the Pong algebra, is related to the most general case of knot Floer homology. Zoltan Szabo illustrated constructions of the algebras \(B(m,k)\), \(C(m,k)\), and \(P(m,k)\) with many explicit examples and explained the structure of modules relevant to construction of homological knot invariants.
Raphael Rouquier posed many questions and outlined a number of directions for studying algebraic structures relevant to 4dimensional topology. In the first part of the talk, he explained that suitable analogues of representations of quantum groups should be 2representations of Lie algebras, and pointed out difficulties with 2tensor products. He then explained that such difficulties disappear in the case of positive part of \(gl(11)\) and, following his joint work with Andrew Manion, outlined how one can recover Heegaard Floer theory from higher representation theory in this special case. In the second part of his talk, Raphael Rouquier raised many interesting questions on what could become of vertex algebras, modular tensor categories, Vassiliev invariants, and the Drinfeld associator upon categorification.
Lisa Piccirillo presented two new constructions of exotic 4manifolds, based on joint work in progress with Adam Levine and Tye Lidman. Both constructions are very explicit and use the same relatively simple cobordisms as building blocks. The first construction relies on a new invariant, derived from Heegaard Floer theory, and leads to exotic 4manifolds with \(b_1 > 0\). One of the new interesting features of this construction is that it produces nondiffeomorphic 4manifolds which are related by knot surgery in the case when the Alexander polynomial is trivial. The second part of the talk was devoted to the construction of simplyconnected exotic 4manifolds with \(b_2 = 10\). Using a symmetry of this construction, one can also obtain exotic examples of definite 4manifolds with \(\pi_1 = \mathbb{Z} / 2\) and \(b_2 = 4\).
On the second day of the conference, Sergei Gukov presented several connections between gauge theory and quantum algebra in the context of multimonopole 4manifold invariants, VafaWitten invariants of 4manifolds, and \(q\)series invariants of 3manifolds. Based on the joint work with Artan Sheshmani and ShingTung Yau, he gave an explicit description of the moduli spaces in VafaWitten theory and in KapustinWitten theory on certain 3manifolds, and summarized several topological applications of the corresponding Floer homology groups. He also conjectured a 2variable generalization of VafaWitten invariants of 4manifolds, based on joint work in progress with PoShen Hsin and Du Pei.
Peter Kronheimer presented results of ongoing work with Tom Mrowka on singular instanton homology for knots and braids with local coefficients. After describing the general algebraic properties of this theory, he explained that, for simple braids (with \(n\) strands) in \(S^1 \times S^2\), the module structure of instanton homology admits an elegant geometric description in terms of a rank1 sheaf on a curve \(C_n\). He presented an explicit form of \(C_n\) for small values of \(n\) and discussed several potential applications to topology and algebraic geometry, in particular to the question of Caporaso, Haris, and Mazur on a uniform upper bound for the arithmetic genus of the rational curves on quintic surfaces.
The program concluded with a talk by Mikhail Khovanov on universal construction of topological theories (a notion weaker than the standard definition of a TQFT) that he illustrated in three types of examples. The first class of examples, based in part on joint works with Radmila Sazdanovic and with Victor Ostrik and Yakov Kononov, illustrated the universal construction in the context of 2dimensional topological theories. Curiously, already in this simple class one finds rich and interesting structures such as Deligne categories. For the second class of examples, Mikhail Khovanov described the combinatorial construction of \(sl(n)\) link homology from the perspective of the universal construction. Specifically, he reviewed the work of LouisHadrien Robert and Emmanuel Wagner on evaluation of closed \(sl(n)\) foams and made contact with the work of Peter Kronheimer and Tom Mrowka on homology of trivalent graphs. In the last part of his talk, Mikhail Khovanov presented yet another class of topological 1dimensional theories associated to a group \(G\) and a finitedimensional representation \(V\) over a field \(k\), that has connections to the theory of pseudocharacters in number theory. This part was based on very recent work with Mee Seong Im and Victor Ostrik.

Agenda
Thursday, March 30th
9:30 AM Aaron Lauda  Heegaard Floer Homology as a quantum invariant 11:00 AM Peter Ozsvath  Constructions in bordered Floer homology 1:00 PM Zoltan Szabo  Knot Floer homology and Pong algebras 2:30 PM Raphael Rouquier  Algebra for 4d invariants 4:00 PM Lisa Piccirillo  Building closed exotica by hand Friday, March 31st
9:30 AM Sergei Gukov  Gauge theory and quantum algebra 11:00 AM Peter Kronheimer  Singular instanton homology for knots and braids: experiments and conjectures 1:00 PM Mikhail Khovanov  Universal construction of topological theories and its applications 
Abstracts & Slides
Aaron Lauda
University of Southern CaliforniaHeegaard Floer Homology as a quantum invariant
View Slides (PDF)Crane and Frenkel’s dream was to categorify the theory of quantum groups and use them to construct new examples of 4dimensional Topological Quantum Field Theories. The program to categorify quantum groups has made a great deal of progress, categorifying quantum groups, tensor powers of irreducibles, Rmatrices, and associated quantum knot invariants. However, the program to construct new 4manifold invariants from even the simples Lie algebra sl(2) has remained elusive.
In this talk I will explain a new strategy for categorifying quantum 4manifold invariants by starting with the very successful theory of Heegaard Floer homology and working backwards to reinterpret its local formulation as categorifying quantum invariants associated to the super Lie algebra gl(11). Having a complete understanding of Heegaard Floer homology within this representation theoretic framework is likely to enhance our understanding of how to develop new 4manifold invariants comings from other Lie algebras.
Peter Ozsvath
Princeton UniversityConstructions in bordered Floer homology
Bordered Floer homology, introduced by Lipshitz, D. Thurston, and me, allows one to express the U=0specialized Heegaard Floer homology of a closed threemanifold Y in terms of a suitable decomposition of Y. This theory should have a generalization to the case of unspecialized Heegaard Floer homology. I will describe aspects of this generalization in the case of threemanifolds with torus boundary. This is joint work with Lipshitz and Thurston.
Zoltan Szabo
Princeton UniversityKnot Floer homology and Pong algebras
In this talk I will describe some recent joint work with Peter Ozsvath on differential graded algebras and knot Floer homology. The bordered Floer homology of Lipshitz, Ozsvath and Thurston is an invariant of threemanifolds with surface boundary, and can be used to study Heegaard Floer homology for closed threemanifolds. Adaptation of these methods, developed in a joint work with Peter Ozsvath, allows effective
computation for a version of knot Floer homology over the ring $$Z/2[U,V]/ (U\cdot V = 0)$$In the talk I will describe some new constructions, some interesting problems and recent advances that are related to extending these methods.
Raphael Rouquier
UCLAAlgebra for 4d invariants
View Slides (PDF)A longstanding problem is to understand the algebraic structures underlying 4d topological field theories and to construct such structures. One issue is to clarify the precise types of field theories that exist, or are expected to exist. I will discuss the case of HeegaardFloer theory. Other questions involve higher dimensional versions of structures arising in 3d.
Lisa Piccirillo
MITBuilding closed exotica by hand
Historically, closed exotic 4manifolds are constructed using cut and paste operations, and their gauge theoretic invariants are computed using gluing formulas. We build closed exotic 4manifolds out of simple 2handle cobordisms, and we compute their Heegaard Floer mixed invariants explicitly using the surgery exact triangle. Exotic examples include definite 4manifolds with \(b_2=4\) and fundamental group \(\mathbb{Z}/2\), and simply connected 4manifolds with \(b_2=10\). We also develop new invariants for 4manifolds with \(b_1\ge 1\), and we use these invariants to show that there are closed exotic manifolds with fundamental group \(\mathbb{Z}\) and homologically essential square zero spheres. We also show that one can change the smooth structure of a closed manifold by performing knot surgery in a fishtail neighborhood using a knot with Alexander polynomial 1. This is joint work in progress with Adam Levine and Tye Lidman.
Sergei Gukov
CaltechGauge theory and quantum algebra
View Slides (PDF)We will discuss gauge theory perspective on qseries invariants of 3manifolds and 4manifolds, as well as aspects that reveal interesting connections to quantum algebra.
Peter Kronheimer
Harvard UniversitySingular instanton homology for knots and braids: experiments and conjectures
View Slides (PDF)I will present some structural results (often experimental) about singular instanton homology, particularly for knots and links in S1 × S2 arising as the closures of spherical braids. There appears to be a potentially rich structure here, in that such a braid gives rise to a rank1 sheaf on an interesting algebraic curve. Possible applications to braided surfaces in Σ × S2, where Σ is a surface with boundary, remain as future targets. This is a progress report on joint work with Tom Mrowka.
Mikhail Khovanov
Columbia UniversityUniversal construction of topological theories and its applications
Foam evaluation gives an explicit algebraic approach to categorification of quantum GL(N) invariants of links. A necessary step in that approach is the universal construction for foam evaluations, giving state spaces that allow to build link homology groups. We review these notions and their applications to GL(N) link homology and to the KronheimerMrowka 3orbifold homology theory. The universal construction is starting to play a role beyond foam evaluation, naturally appearing in Deligne categories and their generalizations, noncommutative power series, automata and pseudorepresentations, and this will be explained in the talk as well.