Seminars in Algebraic Topology

Series

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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.

Craig Westerland: The Stable Homology of the Braid Group with Coefficients Arising from the Hyperelliptic Representation

2nd November, 2023

The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.

Dan Petersen: Moments of Families of Quadratic L-Functions Over Function Fields Via Homotopy Theory

2nd November, 2023

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. (This will be explained in Craig Westerland's lecture on Nov 2.) We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. (This will be explained in my lecture on Nov 3.) Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

Leovigildo Alonso Tarrio: Derivators in additive context

20th October, 2023

By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.

Oscar Randal-Williams: Homeomorphisms of Euclidean Space

6th October, 2023

The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.

Tselil Schramm: Higher-dimensional Expansion of Random Geometric Complexes

25th July, 2023

A graph is said to be a (1-dimensional) expander if the second eigenvalue of its adjacency matrix is bounded away from 1, or almost-equivalently, if it has no sparse vertex cuts. There are several natural ways to generalize the notion of expansion to hypergraphs/simplicial complexes, but one such way is 2-dimensional spectral expansion, in which the local expansion of vertex links/neighborhoods (remarkably) witnesses global expansion. While 1-dimensional expansion is known to be achieved by, e.g., random regular graphs, very few examples of sparse 2-dimensional expanders are known, and at present all are algebraic. It is an open question whether sparse 2-dimensional expanders are natural and "abundant" or "rare." In this talk, we'll give some evidence towards abundance: we show that the set of triangles in a random geometric graph on a high-dimensional sphere yields an expanding simplicial complex of arbitrarily small polynomial degree.

Alina Vdovina: Higher structures in Algebra, Geometry and C*-algebras

23rd June, 2023

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.

Corey Jones: K-theoretic classification of fusion category actions on locally semisimple algebras

14th May, 2023

An action of a tensor category C on an associative algebra A is a linear monoidal functor from C to the monoidal category of A-A bimodules. We consider the problem of classifying (unitary) actions of (unitary) fusion categories on inductive limits of semisimple associative algebras (called locally semisimple algebras). A theorem of Elliot classifies locally semisimple algebras by their ordered K0 groups. We extend this theorem to a K-theoretic classification of fusion category actions on locally semisimple algebras which have an inductive limit decomposition.

Oishee Banerjee: Cohomology and arithmetic of some mapping spaces

10th April, 2023

How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X → Y  from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g., the geometric Batyerv-Manin type conjectures), algebraic geometry (e.g., moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc) and if time permits, homotopy theory (e.g., derived indecomposables of modules over monoids). I will show how one can fruitfully attack such problems by incorporating techniques from topology to the holomorphic/algebraic world (e.g., by constructing a new spectral sequence).