Khyathi Komalan - California Institute of Technology
Title: Non-Hermitian Ribbon Fusion Categories
Abstract: Several open problems have been asked and are being solved on the nature of dagger structures that can be defined on Fusion Categories. Using previous work that has been done investigating the Hermitian Dagger case, we now shift our focus to dealing with Fusion Categories that have a Non-Hermitian Dagger structure on them. What can we do with these categories? Do they yield any interesting properties? We discuss the outline of proofs of theorems on the Müger center, braiding, and spherical structure of such a fusion category, and discuss a potential construction of a TQFT, and what it could mean for us.
Schuchen Mu - Binghamton University
Title: Cyclic Homology and Algebraic K-theory
Abstract: We will use cyclic homology to calculate examples of transfer map on rational algebraic k-theory.
Jack Davidson - University of Sheffield
Title: Homological Stability in a general context
Abstract: Homological stability is a property exhibited by certain families of groups and group homomorphisms between them, which asserts that in a fixed degree, the induced maps on homology become isomorphisms after some stage. The techniques used to prove homological stability for different families of groups all have a common theme, suggesting that a general framework for the proofs may exist. This general framework is given by Randal-Williams and Wahl, who give a framework which encompasses many known examples. This talk will explore the unifying work and how it relates to known examples, as well as providing new ones. (Purely expository - based on Homological Stability for Automorphism Groups, Randal-Williams and Wahl).
Noah Wisdom - Northwestern University
Title: Fields in Equivariant Algebra
Abstract: Tambara functors generalize finite Galois extensions and are of independent interest to equivariant homotopy theorists, appearing as the homotopy groups of ring spectra. Nakaoka defined the notion of a field-like Tambara functor (for which Galois extensions provide an example). In this talk, we'll introduce Tambara functors and classify field-like Tambara functors for the group C_{p^n}. Additionally, in collaboration with Ben Spitz and Jason Schuchardt, we will study what it means for such a Tambara functor to be algebraically closed, or "Nullstellensatzian" (in the sense of Burklund-Schlank-Yuan), with an eye towards an equivariant chromatic Nullstellensatz.
Jackson Morris - University of Washington
Title: Symplectic Orientations
Abstract: Complex orientable cohomology theories possess many nice properties and allow us to better understand the structure of the category of spectra. They have Thom classes for all complex vector bundles and they admit generalized Chern classes, and through the universal such theory we can understand the thick subcategories of Sp. A reasonable question to ask is: to what extent one can generalize this story to motivic homotopy theory? Work of Levine-Morel constructs an analogue of complex orientations in SH(k) and many of the geometric consequences analogize nicely. However, the category if SH(k) has proven to be more sophisticated than that of Sp, and as such a more sophisticated orientation theory is necessary for unpacking its structure. Recent work of Deglise-Fasel has introduced so called symplectic-orientations, with the hope that these are a step in the right direction.
This talk will be an overview of this story. I will start at complex orientations, move to motivic homotopy theory, discuss the extent to which we can use Levine-Morel's GL-orientations, and finally introduce symplectic-orientations.
Zack Garza - University of Georgia
Title: Compactifications of moduli spaces of Enriques surfaces
Abstract: In this talk, I will discuss recent joint work with Alexeev-Engel-Schaffler that provides explicit, combinatorial descriptions of compactifications of the moduli space of numerically polarized Enriques surfaces of degree 2.
We show that the KSBA compactification can be identified with a compactification of a Hodge-theoretic period domain, leveraging Alexeev-Engel’s work on compactifications of moduli spaces of K3 surfaces via integral-affine geometry. We generalize the classical theory of “folding” Dynkin diagrams, and extend Alexeev-Thompson’s theory of toric ADE surfaces to types B and C to classify KSBA-stable degenerations of Enriques surfaces.
Si-Yang - University of Southern California
Title: McKay Correspondence via Symplectic Geometry
Abstract: In this talk, I'll explain the recent paper by McLean and Ritter on proving McKay Correspondence in higher dimensions. The key ingredient of their proof is vanishing of symplectic cohomology, a Floer-theoretic invariant for non-compact symplectic manifolds, and a long exact sequence relating singular cohomology to symplectic cohomology.
Sophia E. Marx - UMass Amherst
Title: 2-Segal Simplicial Sets and Pseudomonoids in Span
Abstract: This (mostly expository) talk is designed as an accessible introduction to the correspondence between 2-Segal simplicial sets, which can be thought of as encoding a kind of weak associativity, and pseudomonoids in the bicategory Span. We’ll also discuss and allude to several natural extensions of this correspondence, adding further structure to our simplicial sets and discovering the kinds of algebraic structures they produce. This talk is based on current work-in-progress (joint with R. Mehta), and builds on previous results by Ivan Contreras, Rajan Mehta, and Walker Stern.
Torgeir Aambo - NTNU
Title: K(n)-local Synthetic Spectra
Abstract: The theory of synthetic spectra has been critical for recent advances in homotopy theory, both theoretical and computational. They have several important features, most notably acting as a deformation of spectra, and as a categorification of the Adams spectral sequence. There is a deformation between K(n)-local spectra and certain derived-complete comodules, but this deformation is not simply the category of synthetic spectra based on K(n). In this talk we will introduce a synthetic version of this deformation, which also categorifies the K(n)-local E-Adams spectral sequence. This is joint work with Marius Nielsen.
Miika Tuominen - University of Virginia
Title: Completions of \Theta_n-spaces
Abstract: The completion functor of Segal spaces developed by Rezk acts as a generalization of the classifying space construction to the $(\infty,1)$-categorical setting, and it can be used to correct the lack of homotopy coherence of naïve nerve constructions. In this talk we will discuss a generalization of Rezk’s completion functor to the $(\infty,n)$-setting of $\Theta_n$-spaces.
Baylee Schutte - University of Aberdeen
Title: Projective Span of Wall Manifold
Abstract: The projective span (pspan) of a smooth manifold is defined to be the maximal number of linearly independent tangent line fields. To initiate a study of projective span, we have calculated the pspan of Wall manifolds, which are certain mapping tori of Dold manifolds. (Classicaly, Wall manifolds were used by C. T. C. Wall in his determination of the oriented cobordism ring). In this talk, we explore how the theory of Clifford algebras and their modules can be used to construct quasi-invariant vector fields on a cover of a Wall manifold, for such vector fields descend to line fields on the quotient. This is joint with Mark Grant and based on arXiv:2311.14107
Jack Romo - University of Leeds
Title: Homotopy Bicategories of (infinity,2)-categories
Abstract: Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models, like Bénabou’s bicategories, tricategories following Gurski and the models of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, including more homotopy-theoretic ones like quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell, Nikolaus and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of $(\infty, n)$-category for general $n$. One might see this as an extension of the fundamental n-groupoid of a homotopy type, a statement I will make precise. In this talk, I will explore current work in the problem of taking homotopy bicategories of non-algebraic $(\infty, 2)$-categories, including a construction of my own. If time permits, I will discuss some of the applications of this problem to topological quantum field theories.