Fall 2026 Algebraic topology seminar

Zhonghui Sun (Columbia Uniersity)

Title: Twisted Bicategorical Shadows and Traces

Abstract: Bicategorical shadows unify cyclicity, Morita invariance, and trace maps in Hochschild homology and THH. Certain equivariant variants, including Cₙ-twisted THH, do not fit the ordinary framework. I will introduce bicategories with G-twisting data and explain how an ordinary shadow induces, for each g ∈ G, a g-twisted shadow on the associated bicategory of G-twists. This construction also realizes twisted Hochschild homology of Green functors and establishes G-Morita invariance. The associated bicategorical trace recovers the twisted Hattori–Stallings trace. For Green functors, the orbitwise traces assemble into a Mackey-functor-valued degree-zero Dennis trace.

Lucy Yang (Columbia Uniersity)
Title: Involutions and the Brauer group in derived algebraic geometry

Abstract: Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) connect the existence of involutions on Azumaya algebras to 2-torsion in the Brauer group. We investigate how the data of an involution on A is reflected in additional structure on its (derived) category of modules and introduce the notion of an involution on a generalized Azumaya algebra. Using the theory of Poincaré ∞-categories developed by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, we introduce involutive versions of the derived Picard and Brauer groups of Toën and Antieau—Gepner and relate them to their classical and non-involutive counterparts. Our invariant can be used to recover the involutive Brauer group of Parimala-Srinivas when the latter is defined; however, they no longer agree even for closed points due to the existence of shifted perfect pairings. We compute these invariants for the sphere spectrum and other examples. As a consequence, we deduce a derived enhancement of a classical theorem of Saltman. This is joint work with Viktor Burghardt and Noah Riggenbach.

David Gepner (Johns Hopkins University)

Title: Higher Categorical Descent

Abstract: Many of the basic notions of homotopy theory lift in an interesting and nontrivial way to the setting of (∞,∞)-categories, provided we replace basic notions like the cartesian product with its dimension-graded analogue, the Gray tensor product. For instance, even the most basic looking higher categories have nontrivial homotopy posets, though the homotopy groups of their groupoidifications are trivial. There are fibrations, long exact sequences, skeletal and Postnikov filtrations, etc. We suggest lax variants of Giraud's axioms which ought to formalize this machinery in much same way that topos theory formalizes these notions in the classical or homotopical settings, allowing for a very general notion of effective descent which seems well suited to applications in mathematical physics.