Fall 2025 Algebraic topology seminar

Abstract: After K(1)-localization, the classical J-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map $\Sigma^{-1}KO^\wedge_2 \to S_{K(1)}$ from the $C_2$-homotopy fixed points (with a twist) to the $\mathbb{Z}_2^\times$-homotopy fixed points of the 2-complete complex topological K-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava E-theory by closed subgroups of the Morava stabilizer group.

We introduce two definitions of the profinite transfer maps. One ad hoc definition is as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map $\Sigma^{-n^2}E_n \to S_{K(n)}$ on homotopy groups is the HFPSS filtration n^2-line. A second definition of the profinite transfer maps is based on the 6-functor formalism for smooth representations of p-adic Lie groups by Heyer—Mann. We prove that the ad hoc and the 6-functor definitions are equivalent.

Abstract: The modern perspective on equivariant stable categories is that they are characterized equivalently by the existence of transfers, duality, and the tom Dieck splitting. The purpose of this talk is to explain an analogous characterization of the G-symmetric monoidal structure when G is finite, and a conjectural picture for what happens when G is an infinite compact Lie group. This is joint work with Mike Mandell.

Abstract: Trace methods are an important tool in understanding K-theory of ring spectra, but usually only work when the rings in question are connective. I will describe joint work with Vova Sosnilo on c-categories, which are stable categories equipped with extra structure that allows many tools for connective rings to apply. Perfect module categories of bounded below rings, as well as perfect complexes on certain stacks, give examples of c-categories. We prove an analog of the Dundas—Goodwillie—McCarthy theorem in this setting. I will then describe how this theory can be used to give formulas for the algebraic K-theory of many categories arising in chromatic homotopy theory in terms of TC.

Abstract: The theory of iterated loop spaces was developed in particular to detect and study generalisaed cohomology theories. In this talk I will review some lesser known applications of operads and iterate loop space theory in the context of braid groups, mapping class groups and commutative K-theory. Motivated by these, I will present a new general construction of infinite loop spaces that allows us to construct infinite loop spaces from objects that may not be expected to give rise to such a structured space, eg Torelli groups.

Abstract: Gromov–Witten invariants in Hermitian K-theory allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field k without assuming that k is the field of complex or real numbers. They were developed in joint work with Kass, Levine, and Solomon in genus 0 for del Pezzo surfaces.
In joint work with Erwan Brugallé and Johannes Rau, we give a complete calculation of these invariants for k-rational del Pezzo surfaces of degree greater than 5. Moreover, we give these invariants the structure of an unramified Witt invariant for any fixed surface and degree. We then construct a multivariable unramified Witt invariant which conjecturally contains all of these invariants for k-rational surfaces. To prove the conjecture for del Pezzo surfaces of degree greater than 5 and obtain the calculation, we study the behavior of these Gromov–Witten invariants during an algebraic analogue of surgery.

Abstract: Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s, Zakharevich defined a "higher" version of scissors congruence, where we don't just ask whether two polytopes are scissors congruent, but also how many scissors congruences there are from one polytope to another.
Zakharevich's definition is a form of algebraic K-theory, which is famously difficult to compute, but I will discuss a surprising result that makes the computation of the higher K-groups possible, at least for low-dimensional geometries. In particular, this gives the homology of the group of interval exchange transformations, and a new proof of Szymik and Wahl's theorem that Thompson's group V is acyclic. Much of this talk is based on joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich, and also with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin Sroka.

Abstract: There is a programme, largely developed by Weiss and Williams, that aims to understand the homotopy type of the diffeomorphism group of a compact, high-dimensional manifold M in terms of Waldhausen's algebraic K-theory of M. In this talk, I will give a brief overview of this programme and present an analogue for spaces of embeddings (of compact manifolds P into M, say). The main difference with the original programme is that the algebraic K-theory of M is replaced by the *relative* algebraic K-theory of the pair (M, M minus P), which, in many cases, coincides with the relative topological cyclic homology of such pair — a far more computable invariant.
As an application, I will report on ongoing joint work with João Lobo Fernandes computing rational homotopy groups of the diffeomorphism group of solid tori S¹ × Dⁿ, n > 4. This follows a strategy of Bustamante–Randal-Williams and extends computations of Budney–Gabai and Watanabe in high-dimensions.

Abstract: Topological Hochschild homology, an invariant of ring spectra, is the realization of a cyclic object defined using Connes' cyclic category and it carries an action of the circle. Real topological Hochschild homology, an invariant of ring spectra with involution, is the realization of a dihedral object defined using the dihedral category and it carries an action of O(2). In this talk, we describe a simultaneous generalization of these constructions, a topological version of homology which takes as input rings with twisted group action, which generalize rings with involution. A new example of interest of this construction is quaternionic topological Hochschild homology, which carries a Pin(2)-action. This is joint work with Gabriel Angelini-Knoll and Maximilien Péroux.

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