Titles and abstracts of plenary talks

Keegan Boyle
Title: Symmetric L-space knots and cyclotomic polynomials
Abstract: I will discuss two questions about polynomials and their interpretations in the context of Alexander polynomials of knots. Both questions deal with a qualitative way in which products of cyclotomic polynomials differ from the general case.

Anthony Conway
Title: Surfaces in 4-manifolds
Abstract: This talk will survey some recent developments in the theory of knotted surfaces. After a brief discussion of classical and high-dimensional knot theory, we’ll focus on dimension 4, outline the differences between smooth and locally flat embeddings, and conclude with some recent unknotting theorems in the topological category.

Philip Hackney
Title: Segal spaces versus partial groups
Abstract: This talk is about a new connection between the d-Segal spaces of Dyckerhoff and Kapranov and the partial groups of Chermak. The higher Segal conditions can be interpreted as a kind of weak higher associativity suitable for cases when compositions/multiplications are not always defined. Higher Segal spaces have applications (when d = 2) in representation theory, geometry, combinatorics, and elsewhere. Meanwhile, partial groups played a key role in Chermak’s proof of the existence and uniqueness of centric linking systems for saturated fusion systems, a major recent result in p-local finite group theory.

We will discuss the d-Segal conditions for simplicial spaces and introduce partial groups from the simplicial perspective. The degree of a partial group is defined to be the least integer k for which it is (2k − 1)-Segal. An ongoing project with Justin Lynd is devoted to computing the degree for interesting classes of partial groups, and in this talk I’ll give some examples and a glimpse at some of the tools we’re using to get a handle on the problem.

Christine Ruey Shan Lee
Title: Shrinking the Khovanov complex for infinite torus links
Abstract: The Khovanov homology for infinite torus links categorifies the Jones-Wenzl projector used to define the colored Jones polynomial, which is closely related to the topological properties of a hyperbolic knot in terms of its boundary slopes and volume. Despite the popularity of Khovanov homology, the structure of the theory remains poorly understood in general, which hinders potential further applica- tions. In this talk, I will introduce and discuss conjectures on the structure of the Khovanov homology of infinite torus links, with a focus on related joint work with Melissa Zhang, Carmen Caprau, Radmila Sazdanovic, and Nicolle Gonzalez.

Lance Edward Miller
Title: Topology and local geometry of the (derived) arc space
Abstract: There is a rich and intimate history, going back to Milnor and Nash, intertwining the topology of the link of and complex analytic germ with an isolated singularity and the singularity information from a log resolution. More recent results of McLean and Budur-Bobadilla-Le-Nguyen deepen these connections to more fully involve arc spaces. In this talk, we'll review this story, its more recent developments, as well as some important tools which give insight into the local structure of arc spaces available in derived algebraic geometry; developed in progress with R. Docampo and C. Eric Overton-Walker. No priori knowledge of arc spaces will be assumed.

Maggie Miller
Title: Branched covers of twist-rolls and spun tori
Abstract: In 2023, Miyazawa produced a family of 4-manifolds with involutions so that each pair is topo- logically equivalent to CP 2 with complex conjugation, but no pair are smoothly equivalent. In 2024, with Hughes and Kim I showed that each manifold is actually diffeomorphic to CP2, so it is the involutions themselves that are exotic. The proof involves understanding a relationship between certain covers of S4 with branch locus a 2-sphere. I’ll talk about those covers and also some new results when the branch locus a related torus, and how it applies to CP2 and also trivializes some homotopy 4-spheres. This is joint with Mark Hughes and Seungwon Kim.

Titles and abstracts of short talks

Yan Tao
Title: Link Floer Stable Homotopy Invariants
Abstract: Manolescu-Sarkar constructed a link Floer stable homotopy type. We show a modified con- struction that is combinatorially friendlier for computing its homotopy invariants, like a Steenrod square.

Omar Dennaoui
Title: Contractibility of the orbit space of a saturated fusion system after Steinberg
Abstract: Recently, Steinberg used discrete Morse theory to give a new proof of a theorem of Symonds that the orbit space of the poset of nontrivial p-subgroups of a finite group is contractible. We extend Steinberg’s argument in two ways, covering more general versions of the theorem that were already known. In particular, following a strategy of Libman, we give a discrete Morse theoretic argument for the con- tractibility of the orbit space of a saturated fusion system.

Kevin Tuttle
Title: From Corks to Contact Structures
Abstract: In Differential Topology, exotic smooth structures begin appearing in dimensions 4 and up, where the equivalence between homeomorphism and diffeomorphism breaks. Dimension 4 specifically is of great note as it is the only dimension with (uncountably many so) exotic R4. In this talk, we will survey the exotic circumstances dimension 4 yields, and talk about a way to chain down the wildness of such phenomenon to lower dimensions via contact structures.