Fall 2026

Title:  What we know about the telescopic categories

Abstract:  The stable homotopy category naturally breaks up into indecomposable building blocks, the telescopic (or T(n)-local) stable categories. These pieces are not well understood in general, rather much attention has been given to their computationally tractable close relatives, the K(n)-local categories. I will give an overview of some of the recent advances in understanding the T(n)-local category, explaining what we understand about its difference with the K(n)-local category, some consequences, and some of the new tools we have for working with the T(n)-local category.

This comes from projects joint with Burklund, Carmeli, Clausen, Hahn, Schlank, and Yanovski.

Title:  Modular Character Sheaves and Monodromic Hecke Categories

Abstract:  Character sheaves provide a geometric framework for studying characters of finite groups of Lie type. Introduced by Lusztig in the 1980s, they played a central role in the classification of irreducible complex representations of these groups and in the computation of their characters. Although the definition of character sheaves can be formulated with modular coefficients, much of Lusztig's characteristic-zero machinery breaks down in this setting.

In this talk, I will give a streamlined overview of the theory of character sheaves using modern language from categorical representation theory. From this perspective, categorical traces allow one to study character sheaves through monodromic Hecke categories. I will describe two models for these categories: a geometric categorification in terms of parity sheaves and an algebraic categorification in terms of Soergel bimodules. A key advantage of these models is that they are tightly controlled by p-Kazhdan–Lusztig combinatorics. This paves the way for a combinatorial approach to the study of modular character sheaves.

Title:  Homotopical first order logic

Abstract:  We will see that model categories also have logical information on their own in the following sense: Given any model category, we can associate to it a class of first-order formulas referring to the fibrant objects of the category. For example, the associated language of the category of small categories, equipped with its canonical model structure, coincides with language for categories defined by Blanc and Freyd, whose central feature is that it respects the equivalence principle.

Similarly, the language we associate to a model category respects the appropriate version of the equivalence principle: two homotopically equivalent objects satisfy the same formulas and replacing parameters by homotopically equivalent ones does not change the validity of a formula.

Finally, if M and N are two Quillen equivalent model categories, their associated languages are, suitably, equivalent.

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Title:  HOMFLY Homology for Oriented Tangles

Abstract:  We introduce an extension of Khovanov and Rozansky's categorification of the HOMFLY polynomial to oriented tangles. Unlike previous such extensions, ours is invariant under the full suite of oriented Reidemeister moves, including the "misoriented" Reidemeister IIb. For oriented links L, this removes the preliminary step of presenting L as a braid closure. Our construction admits an interpretation in terms of Bar-Natan's "canopoly" framework, allowing for local computations. Time permitting, we will briefly discuss potential consequences for functoriality under (non-compact) oriented tangle cobordisms and extracting smooth 4-manifold invariants. Based on forthcoming joint work with William Ballinger.