2026 Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting

Over the past year, the collaboration produced more than 20 papers co-authored by the PIs and organized several focused events. One of the highlights was the summer school followed by the Conference on Modern Developments in Low-Dimensional Topology, held at the ICTP in Trieste. Such events play a crucial role in bringing together young researchers and strengthening the broader community. A monograph featuring the lectures and talks from a similar summer school and conference held in Budapest in 2024 is currently being finalized and is expected to appear in print within the next few months.

At the annual meeting, Director Aaron Lauda gave a talk on the “hot topic” bridging the mathematics of our collaboration and quantum computation. Lauda explained that this connection runs in both directions. On one hand, quantum algebra and quantum topology provide powerful tools applicable to quantum computing. In the other direction, quantum computing can offer faster and more efficient methods for computing knot invariants that are difficult to handle on classical hardware. The key idea is to express Khovanov homology in the language of combinatorial Hodge theory, which reformulates the problem as a study of the spectrum of a Laplacian determined by the knot diagram. Lauda explained that the efficiency of the algorithm can be significantly improved by employing the spanning tree model for Khovanov homology, and he highlighted several interesting open questions. In particular, he observed that analytic torsion surprisingly captures a very large portion of the non-harmonic part of the spectrum.

Peter Ozsváth provided an engaging progress report, based on joint work with Robert Lipshitz and Dylan Thurston, on the development of bordered Floer homology for 3-manifolds with boundary. He focused on the case where the boundary is a torus, deferring the higher-genus case to future work. Ozsváth also presented a pairing theorem that describes how to reconstruct the Heegaard Floer homology of a union from the type A and D modules associated with its pieces. Additionally, he described the structure of the weighted \(A_{\infty}\) algebra associated with the torus and introduced the corresponding graphical calculus, explaining connections to the counting of pseudo-holomorphic disks in the geometric setting.

Tye Lidman and Lisa Piccirillo delivered a joint talk on new applications of alpha-type invariants to the construction of exotic smooth structures on 4-manifolds. Tye opened by introducing a class of 4-manifolds — ideal broken Lefschetz fibrations (ideal BLFs) — which are less rigid than Lefschetz fibrations (symplectic 4-manifolds) yet more constrained than general broken Lefschetz fibrations (which encompass all 4-manifolds). He then gave a general definition of alpha-type invariants: invariants that each 4-dimensional TQFT associates to a 4-manifold and a degree-3 homology class. He explained that these invariants are not only computable for ideal BLFs but also determine whether a 4-manifold admits a genuine Lefschetz fibration over a 2-torus, providing a handle on symplectic 4-manifolds even in cases where Seiberg–Witten invariants vanish.

Lisa Piccirillo continued with applications of alpha-type invariants to ideal BLFs, including new theorems on minimal genus within a given homology class and on the existence of exotic 4-manifolds with prescribed fundamental group, signature, and Euler characteristic. Lisa emphasized that proofs relying on alpha invariants are formal and computationally straightforward, opening avenues for numerous generalizations.

The symplectic geometry of 4-manifolds and Lefschetz fibrations also featured prominently in Zoltán Szabó’s presentation. He described new constructions of exotic smooth structures on 4-manifolds with small Euler characteristics. Motivated by the search for exotic structures on the complex projective plane, he presented an infinite family of smooth fake projective planes with small non-trivial fundamental groups, arising from joint work with András Stipsicz and İnanç Baykur. The main construction begins with a complex projective plane blown up three times and takes the quotient by a free Z/2 action. Notably, the exotic manifolds produced in this way do not admit a symplectic structure. The second part of his talk addressed genus-2 Lefschetz fibrations and the question of the existence of smooth sections.

Raphaël Rouquier gave a forward-looking talk on tensor products and categorified R-matrices in higher representation theory. In particular, he explained that an A-infinity tensor-product construction successfully reproduces the expected tensor-product categories related to Webster’s work, a result of recent joint work with Mark Ebert, a Simons Collaboration postdoctoral fellow and former student of Aaron Lauda. This provided a clear example of the many exchanges fostered by the collaboration. Rouquier also proposed another synergistic connection, namely with work of Gukov and Manolescu via categorified Verma modules. Gukov and Rouquier plan to organize a workshop on this topic.

Francesco Lin presented an interesting new connection between the hyperbolic geometry of 3-manifolds and gauge theory. Specifically, he related monopole Floer homology to the spectral theory of the Dirac operator on the same manifold. His main theorems, developed in joint work with Michael Lipnowski, establish that the existence of sufficiently large spectral gaps implies trivial Floer homology. Francesco explained that variants of the Selberg trace formula provide powerful tools for understanding the relevant spectrum.

Melissa Zhang introduced a new version of Morrison–Walker–Wedrich’s skein lasagna modules, based on joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis. The construction replaces the input balls with neighborhoods of 1-complexes, making the modules more amenable to computations for 4-manifolds with 1-handles. The strategy builds on an isomorphism discovered in earlier joint work with Ian Sullivan, relating the skein lasagna module of S2 x D2 to Rozansky–Willis homology, a version of Khovanov homology for links in connected sums of S2 x S1.

Aaron Lauda
University of Southern California

Quantum Algorithms and the Spectral Geometry of Khovanov Homology
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One of the central challenges in quantum algorithm design is to identify problems of genuine computational interest that admit exponential speedups over classical approaches. Shor’s algorithm for factoring is a landmark example, and quantum algorithms for approximating the Jones polynomial have similarly demonstrated exponential advantage. In this talk, we present a quantum algorithm that estimates the ranks of Khovanov homology groups using Hodge theory. This approach can be applied to a broad class of homology problems given certain assumptions.

In this talk, we present a quantum algorithm developed in collaboration with Alexander Schmidhuber, Michele Reilly, Paolo Zanardi, and Seth Lloyd for estimating the ranks of Khovanov homology groups utilizing a combinatorial Hodge theory. We introduce the notion of “harmonic Khovanov homology,” where we identify unique representatives of homology classes as the kernel of a Khovanov-Hodge Laplacian. While classical computation of these groups scales exponentially with the number of crossings, our quantum algorithm provides an efficient alternative, provided the Laplacian satisfies certain spectral conditions. We will discuss joint work with Jernej Grlj exploring the “higher spectrum” of this Laplacian and the question of what information the non-zero eigenvalues encode about the underlying link diagram. We conclude with numerical evidence for the algorithm’s efficiency and open questions regarding analytic bounds on the spectral gap for general knots.
 

Lisa Piccirillo
The University of Texas at Austin

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Tye Lidman
North Carolina State University

More Exotic Constructions and More Alpha-Type Invariants II

We give some new exotica constructions in dimension 4 by further developing the invariants we studied previously with Levine. This is part two of a joint talk with Lisa Piccirillo.
 

Francesco Lin
Columbia University

Floer Theory and the Geometry of Hyperbolic Three-Manifolds

Floer theory and hyperbolic geometry have revolutionized our understanding of three-dimensional topology in the past few decades. Despite this, uncovering mutual interactions between them (if any) remains an outstanding open problem in the field. In this talk, based on joint work with M. Lipnowski, Francesco Lin will discuss how techniques from spectral theory allow to make some first steps towards this goal.
 

Peter Ozsvath
Princeton University

Bordered Floer Homology for Three-Manifolds with Torus Boundary

Peter Ozsvath will describe on-going work with Robert Lipshitz and Dylan Thurston, in which we define Heegaard Floer invariants for three-manifolds with torus boundary. Ozsvath will also describe a pairing theorem, which describes the Heegaard Floer homology of the torus sum of two such three-manifolds in terms of the bordered invariants of the summands.
 

Lisa Piccirillo
UT Austin

More Exotic Constructions and More Alpha-Type Invariants

We give some new exotica constructions in dimension 4 by further developing the invariants we studied previously with Levine. This is part one of a joint talk with Tye Lidman.
 

Raphaël Rouquier
University of California, Los Angeles

Categorified R-Matrices and Knot Invariants
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2-representations are categorical versions of representations of Lie algebras. Tensoring 2-representations is a homotopical operation of a new type. On the other hand, the braiding is given by a surprisingly simple homological construction. Raphaël Rouquier will explain how the usual braiding or the R-matrix describes some homotopical information, and he will discuss the possibility for these to provide Gukov–Manolescu and Gukov–Pei–Putrov–Vafa invariants.
 

Zoltán Szabó
Princeton University

Exotic Structures on Smooth Four-Manifolds
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The purpose of this talk is to study smooth four-manifolds with small Euler characteristics. Topics will include the construction of fake projective spaces with finite fundamental group, the study of Xiao’s genus-2 Lefschetz fibrations, and some fibration problems for complex ball quotients. Part of the work presented is a joint work with Andras Stipsicz and Inanc Baykur.
 

Melissa Zhang
University of California, Davis

Skein Lasagna Modules with 1-Dimensional Inputs

In this talk, Melissa Zhang will describe joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of Morrison–Walker–Wedrich’s skein lasagna modules by replacing the input balls with neighborhoods of 1-complexes.

This version is more amenable to computations for 4-manifolds with 1-handles. The strategy is to use an isomorphism discovered in previous joint work with Ian Sullivan, where we related the skein lasagna module of S2 x D2 to Rozansky–Willis homology, a version of Khovanov homology for links in connected sums of S2 x S1.

Watch a playlist of all presentations from this meeting here.