Schedule 2024

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. 

Sophia E. Marx - UMass Amherst

 Title: On the correspondence between 2-Segal Simplicial sets and PSM's in Span

Abstract: TBD

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.

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. 

Gabriel Longatto Clemente - Universidade Federal de São Carlos - UFSCar

Title: TBD

Jackson Morris - University of Washington

Title: Symplectic Orientations

Abstract: In stable homotopy theory, there is a notion of a complex-oriented cohomology theory. One can view these theories as those having Thom classes for all complex vector bundles, and one can further see that these spectra receive a map from the complex cobordism spectrum MU. Quillen showed that MU held the universal formal group law, and moreover any complex oriented theory has an associated formal group law related to MU via the map above. The work of Devinatz-Hopkins-Smith showed that understanding this complex orientation data determines much of the global structure of the category of spectra. In motivic homotopy theory, the goal of understanding this global structure is more difficult. In particular, we can run the above game and talk about complex orientations, but it does not give us a complete answer. The question becomes: what type of orientation data can we use to help us here? In this talk, I will talk about the classical theory of complex orientations and their connections with formal group laws, before moving to the motivic case and talking about work on symplectic orientations and formal ternary laws

TBD

Noah Wisdom - Northwestern University

Title: Properties and Examples of A-Landweber Exact Spectra

Abstract: It is classically known that Landweber exact homology theories (complex oriented theories which are completely determined by complex cobordism) admit no nontrivial phantom maps. In this talk, I will propose a definition of $A$-Landweber exact spectra, for $A$ a compact abelian Lie group, and observe that an analogous result on phantom maps holds. I will not prove an equivariant Landweber exact functor theorem, although we will see that $MU_A$, $KU_A$, their $p$-localizations, and $BP_A$ are $A$-Landweber exact through ad-hoc methods. Also, I will explain why a conjecture of May on $KU_G$ is false.

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). 

TBD

TBD

Torgeir Aambo - NTNU

Recording

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

Recording

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

Recording

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

Recording

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.