Titles and abstracts of the main talks
Title: On the affine line
Abstract: Inspired by the pioneering work of Hopkins, Neeman, and Thomason on the classification of thick subcategories of perfect complexes, Balmer constructed for any tensor triangulated category T a geometric object Spc(T) which parametrizes the global structure of T. The subject of tensor triangular geometry is then to study the subtle interplay between Spc(T) and T abstractly as well as for prominent examples.
The goal of this talk is to report on joint work in progress with Schlank and Stevenson which revisits and extends Balmer's framework, by introducing the analogue of affine schemes in the context of (higher) tensor triangular geometry. In particular, we will construct the affine line.
Title: Stratifying the category of modules over cochains on a space with Noetherian mod p cohomology
Abstract: The theory of stratification for triangulated categories with an action of a graded commutative Noetherian ring has been developed in works of Benson-Iyengar-Krause. Within a topological content, the category of module spectra over the ring spectrum of mod p cochains on a topological spaces is stratified when the space is the classifying space of a finite group (Benson-Iyengar-Krause) and a connected compact Lie group (Benson-Greenlees). In recent work, the authors describe a general setting to approach stratification for spaces with Noetherian mod p cohomology. Joint work with T. Barthel, D. Heard, and G. Valenzuela.
Title: Fusion systems, p-completed classifying spaces and the classification of finite simple groups.
Abstract: In the study of saturated fusion systems, previously independent developments in local finite group theory, in modular representation theory and in homotopy theory come together. I will give an introduction to the theory and its applications to different parts of pure mathematics. In particular, I will outline the connections of fusion systems to p-completed classifying spaces of finite groups and to the classification of finite simple groups. Along the way, I will mention some of my own results.
Title: Magnitude and Magnitude homology
Abstract: Magnitude is an invariant of metric spaces, introduced by Tom Leinster, that is analogous (in an appropriate sense) to the Euler characteristic. Magnitude homology, introduced by myself and Willerton, and later extended by Leinster and Shulman, is a homology theory for metric spaces that "categorifies" magnitude in the same sense that Khovanov homology categorifies the Jones polynomial. In my talk I will give a survey of magnitude and magnitude homology, with an emphasis on recent progress in both topics.
Title: Spectral Lie algebras and unstable homotopy theory
Abstract: I will survey recent developments in the theory of spectral Lie algebras. These generalize differential graded Lie algebras to the setting of stable homotopy theory. They allow for generalizations of Quillen’s rational homotopy theory to other localizations arising from chromatic homotopy theory.
Title: Local representation theory and fusion systems.
Abstract: Let G be a finite group and let p be a prime number. The p-local representation theory of G relates three types of structure: representations of G over fields of characteristic zero (character theory); representations over fields of characteristic p (modular representation theory), and embeddings of p-groups in G (fusion systems). I will give an introduction to the interplay between the three themes and will present recent joint work with Linckelmann, Lynd and Semeraro which shows how this interplay leads to a theory of weights for arbitrary fusion systems.
Title: Embedding calculus and diffeomorphism groups
Abstract: Embedding calculus is a technique which studies spaces of embeddings through a tower of successive approximations. For this tower to converge a certain codimension condition needs to be satisfied. Even though diffeomorphisms are embeddings, this codimension restriction fails and hence embedding calculus does not apply to diffeomorphisms. I will explain how one can get around this problem and still extract qualitative information about diffeomorphism groups. Parts of this are joint work with Oscar Randal-Williams or Manuel Krannich.
Title: Character Theory
Abstract: Let G be a finite group. A foundational result of representation theory asserts the vector space of conjugation-invariant functions on G has a canonical basis, given by the characters of the irreducible representations of G. In this talk, I'll discuss some generalizations of this result and describe a general framework in which they can be understood.
Title: Geometry without space: a hyperbolic parable.
Abstract: We revisit ideas that are shared by functional analysts and logicians in order to do geometry without bothering with practicalities such as spaces or dimension.
Our main interest will be the case of hyperbolic geometry and some relations with group representations.
Title: Kazhdan's property (T) and semidefinite programming
Abstract: Kazhdan's property (T) for groups has a number of applications in pure and applied mathematics. It has long been
thought that groups with property (T) are rare among the "naturally-occurring" groups, but it may not be so and it may be
possible to observe this by extensive computer calculations. After an introduction, I will present a computer assisted (but
mathematically rigorous) method of confirming property (T) based on semidefinite programming with some operator
algebraic input. I will report the recent result by M. Kaluba, P. Nowak, and me, and the following result by Kaluba, D. Kielak,
and P. Nowak. It confirms property (T) of Aut(F_d), d>4, which solves a well-known problem in geometric group theory.
Title: On Witt vectors with coefficients
Abstract: We introduce Witt vectors for (non-commutative) rings with coefficients in bimodules. This generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors. We also identify the components of the Hill-Hopkins-Ravanel norm for cyclic p-groups with Witt vectors with coefficients. At the end, applications to characteristic polynomials will be discussed. This is all joint with E. Dotto, A. Krause and T. Nikolaus.
Title: Lie, associative and commutative quasi-isomorphism
Abstract. We present two "Koszul dual" theorems: (A) If two commutative dg algebras in characteristic zero are quasi-isomorphic as dg algebras, then they are also quasi-isomorphic as commutative dg algebras. (B) If two dg Lie algebras in characteristic zero have universal enveloping algebras which are quasi-isomorphic as dg algebras, then the dg Lie algebras are themselves quasi-isomorphic. Theorem B says in particular that two Lie algebras in characteristic zero are isomorphic if and only if their universal enveloping algebras are isomorphic; even this result is completely new. (Joint with R. Campos, D. Robert-Nicoud, F. Wierstra)
Title: Cohomology of Torelli groups
Abstract: It is a basic problem in the cohomology of moduli spaces of Riemann surfaces to describe the cohomology of the Torelli group---the subgroup of the mapping class group of those diffeomorphisms which act trivially on the first cohomology of the surface---as a representation of the symplectic group, at least in a stable range depending on the genus of the surface. This question can be generalised to higher dimensions by replacing the genus g surface with its analogue #^g S^n x S^n. I will present joint work with Alexander Kupers in which we answer this question in dimensions at least 6. Our description is also valid in the classical case 2n=2 assuming a finiteness conjecture about the cohomology of this Torelli group.
Title: Transfer maps in group homology
Abstract: If N is a submanifold of a closed manifold M, there is a transfer map in homology from H_*(M) to H_*(N), going the "wrong way". In this talk, we ask the following a priori wild question: Can one lift the transfer map on the homology of manifolds to one in group homology from H_*(G) to H_*(H) for G and H the fundamental groups of N and M respectively, in a way that is compatible with the transfer map on the level of manifolds? In general certainly no, but for surprisingly many situations this does work, in particular when working with K-theory as (co)homology theory and if the codimension of N in M is at most 3. The talk is based on joint work with Martin Nitsche and Rudolf Zeidler.
Title: Symmetries and deformations: the classification of knots
Abstract: Knots and their groups are a traditional topic of geometric topology. In this talk, I will survey recent developments that originate in coloring invariants and lead to the classification of knots in terms of algebraic structures that conceptualize these invariants (and much more).
Title: Homology of Random Cech complexes on manifolds
Abstract: Given a configuration of n points on a manifold M one might ask how large need one choose r-balls around these points so that the manifold is covered, or the associated Cech complex has homology isomorphic to that of the manifold. Of course this heavily depends on which particular points are given. Thus one is naturally led to the question of what can be expected when the points are chosen at random. We will discuss in particular asymptotic threshold formulae for the radius r depending on n and k so that for larger radii the Cech complex computes the k-th homology of M with probability 1, respectively smaller radii with probability is 0.
This talk is based on joint work with Henry-Louis de Kergorlay and Oliver Vipond building on previous results by Bobrowski, Oliveira, and Weinberger.
Title: Rigidity of periodic cyclic complexes over p-adic integers.
Abstract: Periodic cyclic homology of associative algebras generalizes in many ways DeRham cohomology and more generally crystalline cohomology of algebraic varieties over a field of characteristic zero. Among the properties of De Rham cohomology that can be so generalized are: rigidity under infinitesimal deformations and a regulator map from relative algebraic K theory to relative cyclic homology of a nilpotent ideal (Goodwillie), and the Gauss-Manin connection (Getzler). I will explain how these results generalize to p-adic completions of cyclic complexes over p-adic integers. These generalizations develop recent results of Beilinson and Petrov-Vologodsky.
Title: Graphs, Euler characteristics and zeta functions
Abstract: The rational Euler characteristics of many arithmetic groups can be expressed in terms of values of zeta functions. In a surprising twist, Harer and Zagier showed in 1986 that the same is true for surface mapping class groups. The group Out(F_n) shares many properties with both arithmetic groups and mapping class groups. In 1987 J. Smillie and I found a generating function for the rational Euler characteristic of Out(F_n), using the action of Out(F_n) on a space of graphs known as Outer space. This made computer calculations possible but did not yield qualitative information. I will discuss recent joint work with M. Borinsky that determines the asymptotic behavior of the Euler characteristic and also uncovers a relation with zeta functions.
Title: Topology and arithmetic statistics
Abstract: There are many questions in number theory and arithmetic geometry of the sort “Does the following situation ever occur?” For instance, the inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rationals. Similarly, one might ask whether one expects the rank of elliptic curves to be unbounded.
Arithmetic statistics, broadly speaking, pursues the more quantitative question of how often such situations occur. The extension of the inverse Galois problem to this setting is a conjecture of Malle’s, which predicts an asymptotic formula for the number of occurrences of a given finite group G as the Galois group of a number field, as a function of the discriminant. There are analogous statistical conjectures regarding the distribution of class groups ordered by discriminant (e.g., the Cohen-Lenstra heuristics), or the rank of elliptic curves ordered by height (Goldfeld/Katz-Sarnak).
In this talk, we will give an introduction to these sort of questions, focusing on Malle’s conjecture. Additionally, we will explain how to formulate function field analogues of this conjecture and transform this conjecture into a problem in algebraic topology (about the homology of certain moduli spaces of branched covers of P^1). In joint work with Ellenberg and Tran, we partially solved this problem, giving the upper bound in Malle’s conjecture.
Title: Quantum symmetric spaces from reflection equation and module categories
Symmetric spaces admit rich structures related to Poisson geometry. In this talk we compare two "quantizations" of such structures, either via partial differential equations, or via more algebraic methods. These different models have close ties to the reflection equation (a step beyond the Yang–Baxter equation), and are unified in the framework of module categories. I will explain how such quantization are constructed, and sketch a conjectural correspondence between them.