School of Mathematics

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.

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.

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.

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).

The mathematical work of Vladimir Voevodsky

Dan Grayson
University of Illinois, Urbana-Champaign
September 11, 2018
Abstract: Vladimir Voevodsky was a brilliant mathematician, a Fields Medal
winner, and a faculty member at the Institute for Advanced Study, until his
sudden and unexpected death in 2017 at the age of 51. He had a special flair
for thinking creatively about ways to incorporate topology and homotopy theory
into other fields of mathematics. In this talk for a general audience, I will
sketch his seminal contributions to two broad areas, algebraic geometry and the

What do we mean by "equal"

Pierre Deligne
Professor Emeritus, School of Mathematics
September 11, 2018

Abstract: In the univalent foundation formalism, equality makes sense only between objects of the same type, and is itself a type. We will explain that this is closer to mathematical practice than the Zermelo-Fraenkel notion of equality is.