Topology seminar

University of Minnesota

Location

Mondays at 2:30 in Vincent Hall 570

 Filtrations and derived spaces

  Tyler Lawson , University of Minnesota

 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

 All connective G-spectra come from K-theory

  Maxine Calle, UPenn

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

 Bredon homological stability

JD Quigley, UVA

 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

  Nima Rasekh  , MPIM Title: A univalence maxim for category theory Abstract: This is joint work with Niels van der Weide, Benedikt Ahrens and Paige Randall North. Category theory can now be found all throughout mathematics. This motivates a proper classification of categories and its various generalizations, such as 2-categories and double categories. However, unlike many algebraic structures, such as groups, categories can be studied both up to isomorphisms and equivalences. Moreover, other categorical structures exhibit an even wider range of equivalences, complicating any classification effort.  In this talk I want to explain that in an alternative mathematical foundation, and concretely a univalent one, we can refine our definition of categories and internalize the desired equivalence type. I will in particular apply this perspective to the case of double categories and discuss some implications thereof. No knowledge beyond the definition of a category is assumed for this talk, and everyone is welcome to attend.

Abstract not available

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

Equivariant enumerative geometry

Thomas Brazelton, Harvard University

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

 Polynomiality of surface braid and mapping class group homological representation functors.

 Arthur Soulié  , CNRS

 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

The algebra of categories

  Ralph Kaufmann, Purdue University

 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

(Graph) Configuration Spaces and (Representation) Stability

  Nick Wawrykow, University of Chicago

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

TBD

VH 570 Online Meeting Info: 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

TBD

VH 570 Online Meeting Info: 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

TBD

 Allen Yuan

TBD

VH 570 Online Meeting Info: 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