# Lectures by Faculty

## Towards elementary infinity-toposes

## Voevodsky proof of Milnor and Bloch-Kato conjectures

Abstract:

I will discuss main ideas and steps in the

proof of Milnor and Bloch-Kato Conjectures given by Voevodsky .

## Isotropic motivic category

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 synthetic theory of $\infty$-categories vs the synthetic theory of $\infty$-categories

## Univalent foundations and the equivalence principle

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.

## Galois, Grothendieck and Voevodsky

## On Voevodsky's univalence principle

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.

## A1-algebraic topology : genesis, youth and beyond

## What do we mean by "equal"

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.