Tag - Algebraic topology

Marc Stephan: An equivariant BGG correspondence and applications to free A4-actions

13th June, 2024

A classical question in the theory of transformation groups asks which finite groups can act freely on a product of spheres. For instance, Oliver showed that the alternating group A4 can not act freely on any product of two equidimensional spheres.

I will report on joint projects with Henrik Rüping and Ergün Yalcin and explain that for 'most' dimensions m and n, there is no free A4-action on Sm × Sn and whenever there exists such a free action, then the corresponding cochain complex with mod 2 coefficients is rigid: its equivariant homotopy type only depends on m and n.

This involves an equivariant extension of Carlsson’s BGG correspondence in order to classify perfect complexes over 𝔽2[A4] with 4-dimensional total homology.

Matthew Morrow: Algebraic K-theory and p-adic arithmetic geometry

22nd April, 2024

To any unital, associative ring R one may associate a family of invariants known as its algebraic K-groups. Although they are essentially constructed out of simple linear algebra data over the ring, they see an extraordinary range of information: depending on the ring, its K-groups can be related to zeta functions, corbordisms, algebraic cycles and the Hodge conjecture, elliptic operators, Grothendieck's theory of motives, and so on.

Our understanding of algebraic K-groups, at least as far as they appear in algebraic and arithmetic geometry, has rapidly improved in the past few years. This talk will present some of the fundamentals of the subject and explain why K-groups are related to the ongoing special year in p-adic arithmetic geometry. The intended audience is non-specialists.

Maxime Ramzi: On Endomorphisms of THH

27th March, 2024

Topological Hochschild homology is an important invariant, closely related to algebraic K-theory, and can be seen as a non-commutative analogue of de Rham chains.

In this talk, I will describe various computations of the ring/monoid of endomorphisms of THH in different variants, with an approach based on a generalized version of the Dundas-McCarthy theorem.

Oscar Randal-Williams: Cohomology of moduli spaces: a case study

25th March, 2024

I will explain recent work of Bergström–Diaconu–Petersen–Westerland, and of Miller-Patzt-Petersen-R-W, which uses methods which have been developed over the last 25 years for studying the topology of certain moduli spaces in order to answer a question in arithmetic statistics (the function field analogue of a conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of quadratic L-functions). My focus will be on the translation of this question to a problem in topology, and some of the modern methods which go into solving this problem.

Jay Shah: Real topological Hochschild homology, C2-stable trace theories, and Poincaré cyclic graphs

29th February, 2024

To study topological Hochschild homology as an invariant of stable ∞-categories and endow it with its universal property in this context, Nikolaus introduced the formalism of stable cyclic graphs and trace theories (after Kaledin). On the other hand, Poincaré ∞-categories are a C2-refinement of stable ∞-categories that provide an adequate formalism for studying real and hermitian algebraic K-theory, which should be then well-approximated by the real cyclotomic trace. In this talk, we explain how to systematically provide Poincaré refinements of all the components of Nikolaus's approach to stable trace theories.

Tess Bouis: Motivic Cohomology of Mixed Characteristic Schemes

14th February, 2024

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.

Shachar Carmeli: Cyclotomic Redshift

18th January, 2024

I will discuss a joint work with Ben-Moshe, Schlank, and Yanovski, proving the compatibility of T(n+1)-local algebraic K-theory with the formation of homotopy limits with respect to p-local π-finite group actions on T(n)-local categories. This is a generalization of the results of Thomason for height 0 and Clausen, Mathew, Naumann, and Noel for actions of discrete p-groups in arbitrary chromatic height. I will then discuss the compatibility of K-theory with the chromatic cyclotomic extensions, chromatic Fourier transform, and higher Kummer theory from previous works with Barthel, Schlank, and Yanovski, phenomena we refer to as "cyclotomic redshift''. Finally, I will explain how cyclotomic redshift gives hyperdescent for K-theory along the cyclotomic tower after K(n+1)-localization.

Tom Bachmann: p-adic homotopy theory and E∞-coalgebras

16th December, 2023

I will report on joint work with Robert Burklund. We prove that the canonical functor from p-complete, nilpotent spaces to E∞-coalgebras over the algebraic closure of 𝔽p is fully faithful. This generalizes a theorem of Mandell.

Dan Petersen: Uniform Twisted Homological Stability

3rd November, 2023

Homological stability is now well established as an organizing principle and computational tool in algebraic topology and other areas. In many cases it is of interest to obtain homological stability with twisted coefficients, and the standard choice of such coefficients are the polynomial coefficient systems. All known approaches to homological stability with polynomial coefficients produce a stable range depending on the degree of polynomiality. I will explain a method of obtaining uniform stable ranges for some classes of groups and coefficients of natural interest. This has important consequences in arithmetic statistics, discussed in the number theory seminar on Nov 2.