Topology seminar

University of Minnesota

Location

Mondays at 2:30 in Vincent Hall 570

Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups

Anh Hoang Trong Nam, University of Minnesota Slides

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 heuristics. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle's conjecture over function fields by employing a topological observation which identifies the homology of the braid groups with coefficients arising from braided vector spaces with the cohomology of a quantum shuffle algebra, using the Fox-Neuwirth cellular stratification of configuration spaces of the plane. In this talk, we will extend their techniques to study configuration spaces of the punctured plane and prove a similar result for the homology of the Artin groups of type B. As an application, we will discuss computations when the braid representations are one-dimensional over a field, which shed light on a special case of a conjecture about the homology of mixed braid groups due to Ellenberg-Shusterman.

Vincent Hall 570 - Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9

Exploration of Grothendieck-Teichmueller(GT)-shadows and their action on child's drawings

Vasily Dolgushev, Temple University Recording Slides

The absolute Galois group of the field of rational numbers and the Grothendieck-Teichmueller (GT) group introduced by V. Drinfeld in 1990 are among the most mysterious objects in mathematics. In my talk, I will introduce (the gentle version) of the Grothendieck-Teichmueller group. I will also introduce the groupoid GTSh of GT-shadows that can be used to study this group. I will explain how the groupoid GTSh acts on child's drawings, describe properties of this action and show some interesting examples. My talk is based on papers in preparation with Jacob Guynee, Jessica Radford and Jingfeng Xia.

Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9

Plus constructions and monoid definitions of operad-like structures Abstract: The plus construction was introduced by Baez-Dolan as a means for defining their notion of an opetope and by another name with another intent by Getzler and Kapranov. The plus construction has since proven to be a key component in different operadic theories such as the Feynman categories of Kaufmann and Ward. In this talk, I will discuss joint work with Ralph Kaufmann where we generalize the plus constructions to an endofunctor of symmetric monoidal categories. A special case is given by unique factorization categories whose plus construction yields Feynman categories. As an upshot, we can use this to connect the plus construction to monoid definitions of operad-like structures.

Michael Monaco, Purdue University

Abstract not available

Vincent Hall 570 and via Zoom - Meeting ID: 953 2315 3681 - Passcode: uWgLL9