2024 Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting

The 2024 annual meeting of the Simons Collaboration on New Structures in Low-Dimensional Topology underscored the continuing evolution of research within the discipline, showcasing significant advances and setting the stage for future explorations.

The annual meeting started with a talk by Ciprian Manolescu, who gave an overview of the Rasmussen invariant of knots in a 3-sphere and its generalizations to knots in other 3-manifolds. He started with a very nice introduction into the original version of the Rasmussen invariant (and its generalization to links due to Beliakova-Wehrli and Pardon) and applications to the sliceness of knots. After a brief summary of extensions to higher rank and other variants of Khovanov homology, Manolescu focused on extensions of Khovanov homology to links in $RP^3$ and connected sums of $S^1 \times S^2$. In these homology theories, analogues of the Lee deformation lead to genus bounds of oriented link cobordisms in $I \times RP^3$ and for surfaces in boundary connect sums of $S^1 \times B^3$ and $B^2 \times S^2$, as well as in other particular 4-manifolds with boundary.

Then, Mark Hughes surveyed modern applications of machine learning (ML) to knot theory and low-dimensional topology. After a brief introduction to ML and basics of neural networks, he focused on supervised learning and reinforcement learning. As one of the applications of supervised learning, Mark Hughes described his pioneering and by now well-known work on predicting the slice genus of knots with at most 12 crossings. In the reinforcement learning part, he illustrated how a well-trained agent can exhibit ribonness and quasipositivity of knots whose status was previously unknown. In the end of his talk, Mark Hughes briefly mentioned applications of generative learning and use of ML in automated theorem proving.

The meeting’s focus then shifted towards the algebraic aspects of topology, with Cris Negron and Anna Beliakova exploring the frontiers of quantum groups and algebraic constructions, respectively. Thus, Cris Negron discussed the status of small quantum groups at arbitrary roots of unity. He pointed out that, until 2023, it was unclear what $Rep_q SL(n)_{small}$ (and, more generally, $Rep_q G_{small}$) should be, e.g. when $q = e^{\pi i/3}$, and formulated the problem of defining the category $Rep_q G_{small}$ such that (a) it should be a finite-dimensional (non-semisimple) modular tensor category and (b) there should exist a braided tensor (restriction) functor from $Rep_q G$ to $Rep_q G_{small}$. After stressing that for some $G$ and some roots of unity there does not exist any category satisfying (a) and (b), he presented the main result, the existence and uniqueness theorem for $G$ simply-connected and $q$ a root of unity of even order, such that the order divides the lacing.

Jørgen Ellegaard Andersen gave an overview of resurgence in quantum topology and in quantum Chern-Simons theory. He explained that three seemingly unrelated problems — the asymptotic expansion conjecture, the volume conjecture and the GPPV integrality conjecture — are all naturally tied together in this framework. As one of the main results, Andersen presented a finite-dimensional model (based on finite-dimensional integrals) for the WRT invariants at roots of unity. He also posed a question whether categorification of three different approaches that lead to equivalent modular functors — based on modular tensor categories, conformal field theory and quantization of moduli spaces — is also an equivalence.

Continuing algebraic theme, in her talk Anna Beliakova discussed algebraization of low-dimensional topology, meaning algebraic constructions of categories $nAlg$ for $n=3$ and $4$, such that the cobordism category $3Cob$ is equivalent to $3Alg$ and, similarly, the category of 4-dimensional handlebodies $4HB$ is equivalent to $4Alg$. In fact, as Beliakova explained, the former can be obtained as a corollary to the latter, which in turn is the main result of the talk and is a recent theorem of Beliakova, De Renzi, Bobtcheva and Piergallini (that provides an independent proof to an earlier result of Bobtcheva and Piergallini, who construct $4Alg$ as a category generated by a Hopf algebra object). Beliakova also discussed applications to TQFTs and to the generalized Andrews-Curtis conjecture, reinforcing the role of algebraic methods in advancing the field.

The second day started with a talk by Lev Rozansky about a categorical action of the affine braid group in a setting where one associates a 2-category to a Nakajima quiver variety. Namely, it has two special objects such that the categorical Hom between them enjoys a categorical action of the affine braid group by functors through composition. Rozansky discussed the role of Koszul duality and explained that, in the case of $T^* V$ with a $G$-invariant ‘potential’ function $W$, the category $MF^G (V;W)$ should be viewed as a suitable substitute for the category $DCoh^G (Crit (W))$. This formalism allows to reproduce all known categorical affine braid group actions, including the action on coherent sheaves over Nakajima quiver varieties by R. Anno, on the category of Soergel bimodules, on categories of matrix factorizations by Khovanov-Rozansky, and on derived category of coherent sheaves over the moduli spaces of Hecke correspondences due to Cautis-Kamnitzer. Rozansky explained that the triply-graded HOMFLY-PT link homology also can be realized in this way.

The remaining part of the program was devoted to gauge theory. In his very inspiring and engaging talk, Tomasz Mrowka described a variant of instanton Floer homology where one deals with the fact that gauge transformations do not act freely by considering a connected sum with an atom, a 3-manifold with a link and a bundle that has a unique representation into SU(2). The main theme of the talk revolved around the fact that the instanton homology complex for A # A is an A-infinity algebra, while that for any other admissible 3-manifold is an A-infinity modules over it. Mrowka discussed implications for instanton Floer homology and concluded with a list of open problems.

The program concluded with a talk by Aliakbar Daemi on the $SU(N)$-representation conjecture and its applications to instanton homology of knots and 3-manifolds. For a non-trivial knot $K$ in a homology sphere $Y$, the conjecture postulates the existence of a (particular) non-abelian representation into $SU(N)$. One of the main results presented in this talk was a theorem asserting that the conjecture holds true for $N=3$. (The case of $N=2$ was known before, due to Kronheimer-Mrowka.) Daemi pointed out that extending $U(N)$-instanton Floer homology to sutured manifolds can provide a path to proving the representation conjecture for all integer values of $N>2$, and also discussed applications to instanton homology for knots and 3-manifolds.
The afternoon concluded with an engaging discussion time where participants had the opportunity to further their understanding and forge new collaborations.

Anna Beliakova
University of Zurich

On Algebraisation of Low-Dimensional Topology

The categories of n-cobordisms are among the most studied objects in low dimensional topology. For n=2 we know that 2Cob is a monoidal category freely generated by its commutative Frobenius algebra object: the circle. This result also classifies all TQFT functors on 2Cob. In this talk, I will construct similar algebraic presentations and prove classification results for special categories of 3- and 4-cobordisms. Here, the role of Frobenius algebra is taken by a Hopf algebra. The results are obtained in collaboration with Marco De Renzi, Ivelina Bobtcheva and Riccardo Piergallini.
 

Jørgen Ellegaard Andersen
University of Southern Denmark

Quantum Chern-Simons Theory and Resurgence
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Jørgen Ellegaard Andersen will present a finite dimensional integral approach to Quantum Chern-Simons theory, e.g. we will provide a finite dimensional integral formula for the WRT-invariant. Based on this presentation of the WRT-invariants, Jørgen Ellegaard Andersen will discuss aspects of their resurgence properties.

Aliakbar Daemi
Washington University in St. Louis

Higher Rank Yang-Mills Gauge Theory and Knots

Yang-Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU(n) for higher values of n. In this talk, Aliakbar Daemi will discuss conjectures, questions and a few theorems related to SU(n) Yang-Mills gauge theory, its applications in low dimensional topology and its connection to categorification.
 

Mark Hughes
Brigham Young University

Machine Learning Approaches to Low-Dimensional Topology
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Recent breakthroughs in AI have demonstrated that machine learning algorithms can tackle a range of mathematical problems. These range from solving Olympiad-level geometry problems to guiding the construction of formal proofs in Lean. In this talk, Mark Hughes will focus on applications of machine learning to knot theory. These applications include finding ribbon disks for knots, certifying the quasipositivity of braids, and guiding searches for counterexamples to open problems. The scope of these applications varies from approaches that have already been successfully implemented to more speculative works in progress.
 

Ciprian Manolescu
Stanford University

Generalizations of Rasmussen’s Invariant
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Over the last 20 years, the Rasmussen invariant of knots in S³ has had several interesting applications to questions about surfaces in B⁴. In this talk, Ciprian Manolescu will survey some recent extensions of the invariant to knots in other three-manifolds: in connected sums of S¹ x S² (joint work with Marengon, Sarkar, and Willis), in RP³ (joint work with Willis, and separate work of Chen) and in a general setting (work by Morrison, Walker and Wedrich). Manolescu will describe how these invariants give bounds on the genus of smooth surfaces in 4-manifolds such as CP² – B⁴, S¹ x B³, S² x B², RP³ x I, and the unit disk bundle of S².
 

Tomasz Mrowka
MIT

Prospects for Instantons, Webs and Foams
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This talk with give an overview of applications (actual and potential) for instanton homology for webs and foams.
 

Cris Negron
University of Southern California

Modularity for Quantum Groups at Arbitrary Roots of 1

Cris Negron will discuss constructions of small quantum groups at arbitrary roots of unity. In the end, we associate a finite-dimensional (non-semisimple) modular tensor category to any pairing of a simply-connected reductive group with an even order root of 1. Negron will explain the field theoretic motivations for this work and discuss possibilities for deforming these categories along (generally non-abelian) flat connections.
 

Lev Rozansky
University of North Carolina Chapel Hill

Link Homology from a Stack of D2 Branes

As explained by Gukov, Schwarz and Vafa, the link homology is the space of states of BPS particles in a certain 5-dimensional quantum field theory related to superstrings. These BPS particles can be represented by D2 branes assembled in a stack, whose vibrations are described by a 3d supersymmetric Yang-Mill theory. Its topological B-twist is described mathematically as a 2-category of a particular Nakajima quiver variety. Lev Rozansky will show how this perspective leads to several “coherent” link homology constructions and clarifies their interpretation within 3d topology. This is a joint work with A. Oblomkov and other collaborators.