Lectures by Faculty

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.

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.

Isotropic motivic category

Alexander Vishik
The University of Nottingham
September 12, 2018
Abstract: It was observed for a while (at least, since the times of E.Witt) that the notion
of anisotropy of an algebraic variety (that is, the absence of points of degree prime to a given p on it) plays an important role (most notably, in the theory of quadratic forms).

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.

On Voevodsky's univalence principle

André Joyal
Université du Québec á Montréal
September 11, 2018
Abstract: The discovery of the "univalence principle" is a mark of Voevodsky's genius.
Its importance for type theory cannot be overestimated: it is like the "induction principle" for arithmetic.
I will recall the homotopy interpretation of type theory and the notion of univalent fibration.
I will describe the connection between univalence and descent in higher toposes.