Topology seminar

 Obstruction theory attempts to classify maps X -> Y inductively, using a "look-ahead" to make sure we don't make immediate bad decisions; similarly, spectral sequences use look-ahead to improve techniques for calculation. In this talk I'll aim to describe a technique for trying to analyze a space from a filtration on it, using a similar look-ahead technique.

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

Segal's construction of K-theory turns symmetric monoidal categories into connective spectra, and Thomason later showed that every connective spectrum arises this way (up to equivalence). In the setting of equivariant stable homotopy theory, we can ask what the "correct" notion of equivariant symmetric monoidal category models all connective genuine G-spectra via K-theory. This talk will provide an answer to this question, using the K-theory of categorical Mackey functors. Based on joint work with D. Chan and M. Péroux.

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

 The homology of various sequences of topological spaces often stabilizes. For instance, classical results of McDuff and Segal imply that the homology of unordered configuration spaces of open manifolds stabilizes as the number of points in the configuration increases. In this talk, I will discuss an equivariant analogue of this phenomenon, Bredon homological stability, where homology is replaced by Bredon homology, and spaces are replaced by G-spaces for some finite group G. This is joint work with Eva Belmont and Chase Vogeli. 

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

Abstract not available

VH 570 or Zoom link:  Meeting ID: 943 1752 6327

Classical enumerative geometry asks geometric questions of the form “how many?” and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of “equivariant enumerative geometry”, which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any smooth symmetric cubic surface.

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

 I will describe a general construction of homological representations for families of groups, including classical braid groups, surface braid groups and mapping class groups. This recovers the well-known previous constructions, in particular those of Lawrence and Bigelow, and in this sense it unifies them. The construction is moreover “global” in the sense that, it defines functors on categories whose automorphism groups are the considered families of groups, and which also carries richer structures such as polynomiality. I will thus discuss polynomiality properties of these homological representation functors, and explain their applications, in particular for twisted homological stability. All this represents a joint work with Martin Palmer.

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

 Abstract: We introduce a neat way of regarding categories as bimodule monoids. This facilitates the understanding of many constructions in category theory in terms of representation theory. In particular several versions of monoidal categories, such as PROPs, Feynman categories and unique factorization categories can be understood in this way. It also allows one to single out the special properties of operads and algebras over them. From the representation point of view, the notion of algebra or module is very natural. This lends itself to defining bar.and cobar constructions and gives a natural environment for Koszul duality and curvature. This is joint work with my students Michael Monaco, and Michal Monaco and Yang Mo.

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

How does the topology of a configuration space of points in a space X behave as the number of points increases? McDuff and Segal proved if X is a manifold of dimension at least 2, then the unordered configuration space is homologically stable, i.e., after enough points are added the homology of the configuration space doesn’t change. While this isn’t the quite the case when X is a graph, An—Drummond-Cole—Knudsen showed that there is a reasonable notion of stability that arises from placing points along the edges of X. Building on their work, we prove that the homology groups of the ordered configuration space of a star graph are representation stable in the sense of Church—Ellenberg—Farb, and we use these results to find generators and relations for homology.

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

We present a new approach to sheaf theory for data sets by constructing a Grothendieck topology associated to a Cech closure space. A particularly attractive aspect of this theory is that it applies to many of the major classes of interest to applications: directed and undirected graphs, finite simplicial complexes, and metric spaces decorated with a privileged scale, and on topological spaces, the resulting sheaf cohomology is isomorphic to the usual one. In this talk, we will introduce Cech closure spaces and discuss the construction and its basic properties.

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

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

While the stable homotopy groups of spheres are quite complicated, they are also quite structured. In particular, they contain certain periodic families of elements which can be thought of as breaking up the stable homotopy groups into smaller pieces. Chromatic homotopy theory concerns a categorification of this decomposition, which can be implemented through Bousfield localizations of the category of spectra. There are two such families of localization functors, one with deep connections to geometry (localization at Morava E-theory and K-theory), and the other with excellent categorical properties (localization at a telescope of a type n complex). Ravenel's telescope conjecture asks whether these two families of localization functors are equivalent. In this talk, I will attempt to introduce this circle of ideas and hopefully get around to saying something about how algebraic K-theory allows you to produce specific counterexamples to the telescope conjecture.

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

André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields in a celebrated paper from 1948: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the Betti numbers of the associated complex manifold. We enrich the logarithmic derivative of the zeta function to a power series with coefficients in the Grothendieck--Witt group of stable isomorphism classes of unimodular bilinear forms, using traces of powers of Frobenius in A1-homotopy theory. Building off of work of Morel--Sawant and Bondarko, we construct a symmetric monoidal chain functor from smooth schemes to bounded complexes of homotopy modules. We show the quadratically enriched logarithmic zeta function to be connected to the Betti numbers of the associated real manifold under various restrictions. This is joint work in progress with Tom Bachmann and joint work with Margaret Bilu, Wei Ho, Padma Srinivasan and Isabel Vogt.Location: VH 570 - Online Meeting Info: Zoom link:  https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

 The Bhatt—Scholze theory of prismatic cohomology is a fundamental breakthrough in the study of p-adic cohomology theories in arithmetic geometry.  Recently, Drinfeld and Bhatt—Lurie have introduced stacks which give natural categories of coefficients for prismatic cohomology.  The goal of this talk will be to discuss these stacks from a homotopy theoretic perspective and explain the central role played by the equivariant complex cobordism spectrum.  This is joint work with Devalapurkar, Hahn and Raksit.  

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

 The theory of resolvent degree draws from from Klein's "hypergalois" program and broader industries of reducing numbers of coefficients, having developed into an invariant measuring the complexity of algebraic functions, field extensions, groups, and moduli problems. We offer a concrete introduction to RD through focusing on finite groups, emphasizing the notion of versality paradigmatic of Klein's program. We also discuss the general framework implemented in joint work with Alexander Sutherland and Jesse Wolfson, building on those of Farb–Kisin–Wolfson and Duncan–Reichstein, that permits us to address resolvent questions via classical invariant theory in new ways. We will conclude by reflecting on the rich history of solving polynomials, along with what we do and don't know about resolvent degree.

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

TBD

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