Program Schedule

All talks will take place in Wolfensohn Hall

#### Tuesday, September 11

8:30 a.m. – 9:30 a.m. Registration open

8:30 a.m. - 9:00 a.m.

Morning coffee and tea

9:45 am - 10:00 a.m.

Opening remarks

Robbert Dijkgraaf - Director, Institute for Advanced Study

Video

10:00 a.m. – 11:00 a.m.
 The mathematical work of Vladimir Voevodsky Daniel Grayson - University of Illinois, Urbana-Champaign Video
11:00 a.m. – 11:30 a.m. Coffee break

11:30 a.m. – 12:30 p.m.

What do we mean by "equal"
Pierre Deligne - Institute for Advanced Study

12:45 p.m. – 2:30 p.m. Lunch break
2:30 p.m. – 3:30 p.m.

A1-algebraic topology : genesis, youth and beyond
Fabien Morel - Mathematisches Instit der Universität München

Video

3:30 p.m. – 4:00 p.m. Coffee break

4:00 p.m. – 5:00 p.m.

On Voevodsky's univalence principle
André Joyal - Université du Québec á Montréal

Video

#### Wednesday, September 12

 8:30 a.m. – 9:00 a.m. Morning coffee and tea 9:00 a.m. – 9:30 a.m. Registration open 9:00 a.m. – 10:00 a.m. Galois, Grothendieck and Voevodsky George Shabat - Russian State University for the Humanities Video 10:15 a.m. – 11:15 a.m. Univalent foundations and the equivalence principle Benedikt Ahrens - University of Birmingham 11:30 a.m. – 12:30 p.m. The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories Emily Riehl - Johns Hopkins University 12:45 p.m. – 2:30 p.m. Lunch break 2:30 p.m. – 3:30 p.m. Voevodsky proof of Milnor and Bloch-Kato Conjectures Alexander Merkurjev - University of California, Los Angeles 3:30 p.m. – 4:00 p.m. Coffee break 4:00 p.m. – 5:00 p.m. Isotropic motivic category Alexander Vishik - The University of Nottingham Video

#### Thursday, September 13

 8:30 a.m. – 9:00 a.m. Morning coffee and tea 9:00 a.m. – 9:30 a.m. Registration open 10:00 a.m. – 11:00 a.m. Towards elementary infinity-toposes Michael Shulman - University of San Diego Video 11:00 a.m. – 11:30 a.m. Coffee break 11:30 a.m. – 12:30 p.m. Even spaces and motivic resolutions Michael Hopkins - Harvard University Video 12:45 p.m. – 2:30 p.m. Lunch break 2:30 p.m. – 3:30 p.m. Universal Chow group of 0-cycles and  nilpotence Claire Voisin - Collége de France Video 3:30 p.m. – 4:00 p.m. Coffee break 4:00 p.m. – 5:00 p.m. A search for an algebraic equivalence analogue of motivic theories Eric Friedlander - University of Southern California Video

#### Friday, September 14

 8:30 a.m. – 9:00 a.m. Morning coffee and tea 9:00 a.m. – 9:30 a.m. Registration open 9:00 a.m. – 10:00 a.m. Univalence from a Computer Science Point-of-View Daniel Licata - Wesleyan University 10:15 a.m. – 11:15 a.m. Algebraic K-theory, combinatorial K-theory and geometry Inna Zakharevich - Cornell University 11:30 a.m. – 12:30 p.m. On the proof of the conservativity conjecture Joseph Ayoub - University of Zurich Video 12:45 p.m. – 2:30 p.m. Lunch break 2:30 p.m. – 3:30 p.m. Perverse schobers and semi-orthogonal decompositions Mikhail Kapranov - Institute for Advanced Study 3:30 p.m. – 4:00 p.m. Coffee break 5:30 p.m. - 7:30 p.m. Conference dinner  Program

The workshop is free of charge and attendees must register to attend the workshop.

Invited Speakers

Below is a summary of the speaker talks that will take place during the workshop.

Benedikt Ahrens - University of Birmingham

Wednesday from 10:15 a.m. - 11:15 a.m.

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.

#### Joseph Ayoub - University of Zurich

Friday from 11:30 a.m. - 12:30 p.m.

On the proof of the conservativity conjecture.

Abstract: I will review the strategy of the proof of the conservativity conjecture for the classical realisations of Voevodsky motives over a characteristic zero fields.  I will also mention some other consequences of this proof such as the nilpotence of endomorphisms acting by zero on cohomology.

#### Pierre Deligne - Institute for Advanced Study

Tuesday from 11:30 a.m. - 12:30 p.m.

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.

#### Eric Friedlander - University of  Southern California

Thursday from 4:00 p.m. to 5:00 p.m.

A search for an algebraic equivalence analogue of motivic theories.

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.  Over the intervening years, Haesemeyer, Walker, and I have sought to formulate such a theory over more general fields, a theory in which algebraic equivalence replaces the role played by rational equivalence in motivic cohomology and algebraic K-theory.  This talk will report on challenges and conjectures.

#### Daniel Grayson - University of Illinois, Urbana-Champaign

Tuesday from 10:00 am - 11:00 am

The mathematical work of Vladimir Voevodsky

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 foundations of mathematics.  A colleague commented about his work in the former area, which deals with polynomial equations in various alternative number systems, that if mathematics were music, then Voevodsky would be a musician who invented his own key to play in.  His work in the latter area has led to a new alternative foundation for all of mathematics, opening up a new landscape populated by fundamental objects unseen in the traditional foundation provided by set theory, and in which the notion of equality is interpreted in an unexpected way.  It also hastens the day when our mathematical literature has been verified mechanically and referees are relieved of the tedium of checking the proofs in articles submitted for publication.

#### Michael Hopkins - Harvard University

Thursday from 11:30 a.m. - 12:30 p.m.

Even spaces and motivic resolutions

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.

#### André Joyal - Université du Québec á Montréal

Tuesday from 4:00 p.m. - 5:00 p.m.

On Voevodsky's univalence principle.

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. I will sketch applications (direct and indirect) to homotopy theory (Blakers-Massey theorem, Goodwillie's calculus) and to higher topos theory (higher sheaves) [joint work with Mathieu Anel, Georg Biedermann and Eric Finster].

#### Mikhail Kapranov - Institute for Advanced Study

Friday from 2:30 p.m. to 3:30 p.m.

Perverse schobers and semi-orthogonal decompositions.

Abstract: Perverse schobers are conjectural "perverse sheaves of triangulated categories". This concept can be made precise when the base is a Riemann surface, in which case it can be realized as a certain system of triangulated categories and semi-orthogonal decompositions (recollements). One has the  corresponding categorification of cohomology  which can be seen as a kind of Fukaya category with coefficients. Schobers on surfaces can be also seen as describing a categorification of Picard-Lefschetz theory.  Because of this, this formalism is related to the "algebra of the infrared" of Gaiotto-Moore-Witten.

The talk is based on joint works (some in progress)

with A. Bondal, T. Dyckerhoff, V. Schechtman and Y. Soibelman.

#### Daniel Licata - Wesleyan University

Friday from 9:00 a.m. to 10:00 a.m.

Univalence from a Computer Science Point-of-View

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.  Voevodsky's introduction of the univalence axiom was thus exciting from a computer science point view, because it suggested new programs that could be added to type theory, and new principles that could be used in program verification.  However, for such applications, it is necessary to know that the addition of univalence preserves the computational character of type theory, which is Voevodsky's homotopy canonicity conjecture.  While homotopy canonicity has not yet been proved as stated, this problem has led to a great deal of research aimed at understanding the computational meaning of univalence.  This has resulted in several new constructive models of univalent foundations, and new type theories and programming languages based on them.  In this talk, I will give a high-level survey of the motivations for, challenges of, current knowledge about, and applications of work on this problem.

#### Alexander Merkurjev - University of California, Los Angeles

Wednesday from 2:30 p.m. - 3:30 p.m.

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.

#### Fabien Morel - Mathematisches Institut der Universität München

Tuesday from 2:30 p.m. to 3:30 p.m.

A1-algebraic topology : genesis, youth and beyond

Abstract: This talk will be a survey on the development of A1-homotopy theory, from its genesis, and my meeting with Vladimir, to its first successes, to more recent achievements and to some remaining open problems and potential developments.

#### Emily Riehl - Johns Hopkins University

Wednesday from 11:30 a.m. to 12:30 p.m.

The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories

Abstract:  Homotopy type theory provides a “synthetic” framework that is suitable for developing the theory of mathematical objects with natively homotopical content. A famous example is given by (∞,1)-categories — aka “∞-categories” — which are categories given by a collection of objects, a homotopy type of arrows between each pair, and a weak composition law. In this talk we’ll compare two “synthetic” developments of the theory of ∞-categories — the first (joint with Verity) using 2-category theory and the second (joint with Shulman) using a simplicial augmentation of homotopy type theory due to Shulman — by considering in parallel their treatment of the theory of adjunctions between ∞-categories.

#### George Shabat - Russian State University for the Humanities

Wednesday from 9:00 a.m. to 10:00 a.m.

Galois, Grothendieck and Voevodsky

Abstract:  The talk will start with discussing the common features of the three mathematicians from the title: their non-standard education and specific relations with the community, outstanding imagination, productivity and contribution to the mathematics of future. The main part will be devoted to the development of the Grothendieck ideas (mostly from "Esquisse d'un Programme") by Voevodsky, including the application of dessins d'enfants to the inverse Galois problem. Finally, some problems of univalent foundations will be mentioned in the context of dessins-labelled stratifications of moduli spaces of curves.

#### Michael Shulman - University of San Diego

Thursday from 10:00 a.m. - 11:00 a.m.

Towards elementary infinity-toposes

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.

Recently, Grothendieck's toposes of sheaves have been generalized to infinity-toposes of stacks.  The internal language of such infinity-toposes is expected to be a form of Homotopy Type Theory, with central contributions from Voeodsky's Univalent Foundations program.  However, an "elementary" notion of infinity-topos analogous to Lawvere and Tierney's elementary toposes has been lacking.  I will give an introduction to this problem and survey recent progress on it.

#### Alexander Vishik - The University of Nottingham

Wednesday from 4:00 p.m. - 5:00 p.m.

Isotropic motivic category

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). Taken to the limit, this leads to the idea of “Isotropic motivic category”. This category is obtained from the motivic category of Voevodsky by, roughly speaking, killing the motives of all varieties anisotropic over the ground field. In practical terms, this is achieved by applying the projector on DM(k) coming from Cech simplicial schemes. The latter are “forms” of a point and measure how far a given variety is from having a zero cycle of degree 1. These simplicial schemes were heavily used by Vladimir Voevodsky in the proofs of Milnor and Bloch-Kato conjectures. The resulting category, introduced originally for the study of the Picard group of Voevodsky’s category, appears to be related to: numerical equivalence with finite coefficients, Milnor’s operations, and other interesting topics. It can serve as just another illustration of what can be done using the tools provided to us by Vladimir.

#### Claire Voisin - Collége de France

Thursday 2:30 p.m. - 3:30 p.m.

Universal Chow group of 0-cycles and  nilpotence

#### Inna Zakharevich - Cornell University

Friday from 10:15 a.m. - 11:15 a.m.

Algebraic K-theory, combinatorial K-theory and geometry

Abstract:  One way of studying varieties has been through additive invariants: maps $\mu:\{\hbox{varieties}\} \to A$, where $A$ is an abelian group and for any closed subvariety $Y$ of $X$, $\mu(X) = \mu(Y) + \mu(X\backslash Y)$.  Often such measures take values in a group $A$ which is $K_0(\mathcal{E})$ for some exact category $\mathcal{E}$.  The category of varieties itself often behaves like an exact category, and can be assigned its own $K$-theory, in which $K_0$ is exactly the Grothendieck ring of varieties: the free abelian group generated by varieties under the relation that $[X] = [Y] + [X \backslash Y]$ for a subvariety $Y$ of $X$.  Thus an additive invariant is a homomorphism between $K_0$-groups, which begs the question: can such invariants be lifted to become maps between $K$-theory spectra?  In this talk we will discuss the construction of the $K$-theory of varieties and a method for lifting motivic measures.  As an application of this method we lift both the Euler characteristic of a variety (for varieties over $\mathbf{C}$) and the local zeta function of a variety (for varieties over finite fields).  this is joint work with Jonathan Campbell and Jesse Wolfson.