Topology seminar

 In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle’s conjecture for function fields by studying stability of the homology of braid groups with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study character sums of the resultant of monic square-free polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman.

VH 570  Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

It is well known that the homeomorphism group of a disk relative to its boundary is contractible.  This is known as the Alexander trick, and was published 100 years ago.  I will discuss joint work with Randal-Williams on the homeomorphism group of a compact contractible manifold relative to its (not necessarily simply connected) boundary, which we prove to be contractible if the dimension is at least 6.

 VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

The quest we have started with Hisham Sati for physically motivated structures on certain loop spaces continues. In this talk I will describe our latest finds: the identification of the rational homotopy type (represented by a minimal model) of the k-fold free loop space divided by the k-torus T^k action, Map(T^k, S^4)//T^k and finding the E_k-symmetry of that minimal model, where E_k stands for the exceptional series of simple Lie algebras.Location: VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

 "Classically", Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann has garnered interest in related structures which appear in motivic homotopy theory -- these Motivic Mackey Functors and Motivic Tambara Functors do not have anything to do with group-equivariance, but have the same axiomatic. In this talk, I'll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions. The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I'll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context.

VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

Classical resolvent problems (essential dimension, essential p-dimension, resolvent degree, . . .) ask some form of "How complex is . . . a polynomial, an enumerative problem, a branched cover, a variation of Hodge structure, . . .?" An idea going back to Arnold is that characteristic classes should be able to detect this intrinsic complexity. However, to make this work one must show that the relevant characteristic class remains nonzero under restriction to arbitrary Zariski open subvarieties. In this talk, we describe a new method for solving this restriction problem in many cases using prismatic cohomology. As an application, we prove a conjecture of Brosnan that for a complex abelian variety A, the essential p-dimension of the p-isogeny cover A\to A equals dim A for all but finitely many p. This is joint work with Benson Farb and Mark Kisin.

VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

I will describe a categorical framework for homological algebra which simultaneously generalizes categories of R-modules (moreover exact categories) and the category of finite sets (moreover extensive categories). The homological theory of chain complexes of finite sets provides a very simple model of homological concepts equipped with visualization tools that may provide helpful insights for more general homological algebra. With a few minor adjustments, this framework also serves as a setting for generalized algebraic K-theory, extending several classical K-theory theorems to more combinatorial objects such as finite sets and algebraic varieties, resulting in a chain complex model for the K-theory of finite sets (and moreover any extensive category). Based on joint work with Maru Sarazola.Location: VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

The Algebraic K-theory of a smooth manifold is a receptacle for many sensitive invariants. The driving example is the classical fact that the H-cobordism type of a manifold is completely controlled by only the fundamental group of its Algebraic K-Theory space. In fact, there is a reduction to a related invariant that only depends on the fundamental group of the manifold M. Turning to higher invariants, we are not so lucky, but there is still a comparison map called the linearization map that lets us compute parts of the Algebraic K-theory space in good situations. This talk will discuss a new construction of the linearization map when we are presented with a manifold together with the action of a finite group. If time permits, we will also discuss future directions with a view towards equivariant stable cobordism. These results are joint with D. Chan and M. Calle.

VH 570  Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

Let C be a stable $\infty$-category. A Poincaré structure on C is a contravariant quadratic functor out of C into spectra, which satisfies a non-degeneracy condition and provides a duality $C^{op}\xrightarrow{\simeq} C$. When applied to module categories of rings, among other things, this encodes involutions of the base ring. In this talk we want to venture into the world of Poincaré $\infty$-categories, which Is due to Lurie, and explore what it can say about Brauer groups. This is joint work with Ben Antieau, Noah Riggenbach and Lucy Yang.

VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

 The free loop space of a topological space has a canonical circle action given by rotating loops, making it an S^1-space. The work of Jones, Goodwillie, and others, relates the equivariant homology of the free loop space to the cyclic homology of the algebra of singular chains on the topological monoid of based loops. One can model the free loop space in terms of the chains on the underlying space considered as a categorical coalgebra, a notion Koszul dual to a non-negatively graded dg category. This construction is "as small as possible", has no hypotheses on the underlying space and is suitable for computations in (non-simply connected) string topology. I will present a cyclic theory for categorical coalgebras and dg-categories extending the theory of cyclic homology for (dg) algebras and coalgebras. In particular, the cyclic chains of the categorical coalgebra of chains on a simplicial set provides a model for the S^1-equivariant chains on the free loop space that is suitable for computations. The proofs of these results can be understood in terms of a combinatorial model for the unit of the Bar-Cobar adjunction.

VH 570 Online Meeting Info: Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

Strict n-categories can be defined inductively as categories enriched in strict (n-1)-categories, starting with sets when n=0. Similarly, $(\infty,n)$-categories can be described as $\infty$-categories enriched in $(\infty,n-1)$-categories, starting with spaces when n=0. I will describe an extension of this to more general enrichments: if V is an $E_n$-monoidal $\infty$-category, then V-enriched $(\infty,n)$-categories can be defined both by iterated enrichment starting with V and as a V-enriched version of Barwick’s n-fold Segal spaces. Time permitting, I will attempt to explain how this follows quite formally from $(infty,2)$-categorical manipulations of fibrations, lax transformations and Gray tensor products.

VH 570Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

I will discuss recent joint work with Tom Bachmann and Mike Hopkins wherein we establish a version of the Freudenthal suspension theorem in motivic homotopy theory.  Along the way, I will try to give a gentle introduction to motivic homotopy theory and explain why one might be interested in establishing such a result.Location: VH 570 Online Meeting Info: Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

TBD

 VH 570  Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

 We describe a new homological stability result for a generalized version of Hurwitz spaces. This builds on previous work of Ellenberg-Venkatesh-Westerland, showing that homology groups of certain Hurwitz spaces stabilize. We generalize this in two directions. First, we work with covers of arbitrary punctured Riemann surfaces instead of just the disc. Second, we generalize the result to "coefficient systems," which are essentially a sequence of compatible local systems on configurations spaces. After detailing the above homological stability result, we will then explain how both these generalizations are employed to prove versions of numerous conjectures from number theory relating to the distributions of ranks of elliptic curves and Selmer groups of elliptic curves.

VH 570 Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327