2025 Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting

Mike Miller Eismeier
University of Vermont

ASD Connections & Cosmetic Surgery
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The cosmetic surgery conjecture predicts that, given a knot K in a 3-manifold, the oriented diffeomorphism type of surgery on K determines the surgery slope (up to oriented diffeomorphism). For knots in the 3-sphere, a sequence of restrictions coming from Heegaard Floer homology implies that if the cosmetic surgery is false for K, then r-surgery on K is not oriented diffeomorphic to (-r)-surgery, for some r in {2, 1, 1/2, 1/3, …}, but Heegaard Floer homology techniques reach a limit here.

Mike Miller Eismeier will discuss how a quantitative enhancement of instanton homology rules out the cases r = 1/n, leaving only the possibility of 2-surgery. Miller Eismeier will discuss limitations of this approach as well as possible future developments.
 

Sergei Gukov
Caltech

AI and AC
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An alternative title for this talk could be “Learning Hardness.” While it comes as no surprise that solving challenging research-level math problems drives progress in mathematics, it is far less obvious that solving such long-standing open problems also plays a crucial role in shaping the next generation of AI systems. We live in an exciting time where mathematics and AI can greatly benefit each other, and the goal of the talk is to explain how and why, drawing on specific examples from knot theory and combinatorial group theory. The notion of “hardness” — as in hard ribbon knots or hard AC presentations — plays a central role.

This talk is largely based on recent work with A.Shehper, A.Medina-Mardones, L.Fagan, B.Lewandowski, A.Gruen, Y.Qiu, P.Kucharski, and Z.Wang.
 

Gary Matthew Guth
Stanford University

Real Heegaard Floer Homology

There has been a burst of interest in gauge theoretic invariants of 3- and 4-manifolds equipped with an involution, developed in various contexts by Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Notably, Miyazawa proved the existence of an infinite family of exotic RP^2-knots using real Seiberg-Witten theory. In joint work with Ciprian Manolescu, we construct an invariant of based 3-manifolds with an involution, called real Heegaard Floer homology. This is the analogue of Li’s real monopole Floer homology. Our construction is a particular case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. In the case of Heegaard Floer homology, the construction starts from a Heegaard diagram where the involution swaps the alpha and beta curves. We prove that real Heegaard Floer homology is indeed a topological invariant of the underlying pointed real 3-manifold. Further, we study the Euler characteristic of our theory, which is the Heegaard Floer analogue of Miyazawa’s invariant for twist-spun 2-knots. This quantity is algorithmically computable and, indeed, appears to agree with Miyazawa’s invariant.
 

Peter Kronheimer
Harvard University

SO(3) Versus SU(3) in the Instanton Homology for Webs and Foams
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In joint work with Tom Mrowka, an instanton homology for webs and foams was constructed previously using SO(3) gauge theory. Among other motivations, there were close connections with the foam evaluations in subsequent work of Khovanov-Robert for example. It turns out that the gauge group SO(3) can be replaced by SU(3) with surprisingly little change. A non-vanishing theorem for the SO(3) theory no longer holds, but other results carry over, and the structure simplifies. This talk will describe the relationship.
 

Lisa Piccirillo
Univeraity of Texas at Austin

Exotic Phenomena in Dimension 4

In favorable circumstances, topological 4-manifolds can be classified. In contrast, smooth 4-manifolds remain poorly understood. Several hard questions in 4-manifold topology boil down to asking whether there can be distinct smooth (sub)manifolds with the same topological type; such manifolds are called exotic. In this talk, Lisa Piccirillo will discuss a few methods to demonstrate exotic phenomena and comment on how these methods might be adapted towards addressing some of these questions.
 

Lev Rozansky
University of North Carolina Chapel Hill

Quiver-Like Varieties for the Geometric Categorification of Super-Algebras Gl(M|N) and the Associated Link Invariants

We show how a string-theory inspired reinterpretation of the Hamiltonian reduction involved in the definition of the Nakajima quiver varieties leads to a wider class of varieties, some of which should be associated with the super-algebras gl(m|n).

This is a joint project in progress with A. Oblomkov and Li Han.
 

Andras Stipsicz
Alfréd Rényi Institute of Mathematics

Smooth Structures on Closed Four-Manifolds with Non-Trivial Fundamental Group
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It is a key question of four-dimensional topology to determine which closed topological four-manifolds admit more than one (or even, infinitely many) smooth structures.

In the lecture, we collect several constructions for producing ‘exotic’ smooth structures, which lead to a satisfactory answer in some cases with non-trivial fundamental group. We also examine the potential source of exoticness through the genus function on the second homology group.
 

Paul Wedrich
University of Hamburg

From Link Homology to Topological Quantum Field Theories
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Quantum topology first revealed that the Jones polynomial—and many other knot and link invariants—originate from braided monoidal categories of quantum group representations, providing a foundation for associated 3- and 4-dimensional topological quantum field theories (TQFTs).

Khovanov’s categorification of the Jones polynomial suggests an analogous higher categorical structure, hinting at connections to 4- and 5-dimensional TQFTs via braided monoidal 2-categories. In this talk, I will outline four types of TQFTs emerging from link homology—4d and 5d, linear and derived—and survey the current landscape of concrete examples.