School of Mathematics

Univalence from a computer science point-of-view

Dan Licata
Wesleyan University
September 14, 2018
Abstract: One formal system for Voevodsky's univalent foundations is Martin-Löf's type theory. This type theory is the basis of proof assistants, such as Agda, Coq, and NuPRL, that are used not only for the formalization of mathematics, but in computer science for verification of programs, systems, and programming language designs and implementations. These applications rely on the fact that constructions in type theory can be interpreted constructively as programs.

Towards elementary infinity-toposes

Michael Shulman
University of San Diego
September 13, 2018
Abstract: Toposes were invented by Grothendieck to abstract properties of categories of sheaves, but soon Lawvere and Tierney realized that the elementary (i.e. "finitary" or first-order) properties satisfied by Grothendieck's toposes were precisely those characterizing a "generalized category of sets". An elementary topos shares most of the properties of Grothendieck's, as well as supporting an "internal language" that enables it to be used as a basis for mathematics.

Even spaces and motivic resolutions

Michael Hopkins
Harvard University
September 13, 2018
Abstract: In 1973 Steve Wilson proved the remarkable theorem that the even spaces in the loop spectrum for complex cobordism have cell decompositions with only even dimensional cells. The (conjectural) analogue of this in motivic homotopy theory leads to a surprising resolution of cellular varieties into motivic complexes. In this talk I will survey this situation and describe joint with with Mike Hill and with Jean Fasel and Aravind Asok progress and on potential applications.

A search for an algebraic equivalence analogue of motivic theories

Eric Friedlander
University of Southern California
September 13, 2018
Abstract: We reflect on mathematical efforts made years ago, initiated by Blaine Lawson and much influenced by Vladimir Voevodsky's work. In work with Lawson, Mazur, Walker, Suslin, and Haesemyer, a "semi-topological theory" for cohomology and K-theory of complex (or real) varieties has evolved which has led to a few computations and many conjectures.

Univalent foundations and the equivalence principle

Benedikt Ahrens
University of Birmingham
September 12, 2018

Abstract: The "equivalence principle" says that meaningful statements in mathematics should be invariant under the appropriate notion of equivalence of the objects under consideration. In set-theoretic foundations, the EP is not enforced; e.g., the statement "1 ϵ Nat" is not invariant under isomorphism of sets. In univalent foundations, on the other hand, the equivalence principle has been proved for many mathematical structures. In this introductory talk, I give an overview of univalent foundations and the equivalence principle therein.