Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
In this talk we will examine a result (due to Robin Cockett and Richard Garner) identifying Cockett and Stephen Lack's notion of restriction category as a certain class of 2-categories weakly enriched over a particular base, constructed via Day convolution in a weak double category. We then suggest some examples from differential geometry, functional and complex analysis, and computability theory which call for a generalization of the restriction category concept, and a strategy for extending Cockett and Garner's structure theory result for ordinary restriction categories to this new setting.In this talk, we provide a general overview of a few concepts in computability theory which have motivated our study of restriction categories and related structures: partial combinatory algebras, their so-called functional completeness, and their ability to generate categories known as realizability toposes, with structures known as triposes as an intriguing intermediate step in one account of how realizability toposes may be constructed. We then introduce Turing categories as a certain categorification of partial combinatory algebras based on restriction categories, and illustrate how some of previous concepts translate into this new setting.This talk is the first in a series which reviews the seminal work of Soren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss. The main result is a refinement of the Pontryagin-Thom equivalence to a space level equivalence, going between the classifying space of the cobordism category and a certain spectrum. The authors use this equivalence to prove a conjecture by David Mumford about the cohomology of the mapping class group of Riemann surfaces. In this first talk, I will begin motivation and review the classical Pontryagin-Thom construction. I will then describe a categorification of the collapse map, which is claimed to induce an equivalence at the level of classifying spaces. Various bordism categories will be introduced, all of which will be equivalent. If there is time, I will introduce the Madsen-Tillmann spectrum.In the second installment of this series, I will introduce the topological bordism category and discuss its classifying space. Then I will review some basics of sheaf theory and discuss the geometric realization (or shape) of sheaves on manifolds. A sheaf variant of the bordism category will be introduced and it will be shown (next time) that its geometric realization is precisely the classifying space of the bordism category.In this talk, I will introduce the Madsen–Tillmann spectrum and discuss its connection with the Thom spectrum. I will prove the first of the two main theorems that imply Mumford's conjecture.In this talk, I will discuss an equivalence between the category of etale groupoids internal to locales and a certain subcategory of inverse semigroups. This generalizes the well-known equivalence of pseudogroups and effective etale Lie groupoids, as well as the correspondence between etale groupoids and quantales, due to Pedro Resende and Lawson–Lenz.We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. Resolving each crossing in a knot diagram produces 2 new knot diagrams; knot diagrams with over 5 crossings become increasingly difficult to work with. In this talk, I will introduce a method for automating these computations and discuss how this method was implemented as a Python program. I will illustrate the use of the program in several known examples, demonstrating how examples obtained through several months of work can be can now be obtained in less than 5 minutes. This program will be used to generate a library of examples for testing various conjectures in Chern-Simons theory.