Characteristic classes in stable motivic homotopy theory pt.2 | Frédéric Déglise, CNRS, ENS Lyon

Characteristic classes in stable motivic homotopy theory, part 2
Speaker: Frédéric Déglise, CNRS, ENS Lyon

Lecture notes 2
Problem set 2

Abstract : Characteristic classes is a pillar of differential geometry. The book of Milnor and Stasheff was a corner stone and the work of Quillen on formal group laws and cobordism showed their fundamental importance in stable homotopy,giving rise to entirely new subject such as chromatic homotopy theory.

It is natural to look for an analogous picture in stable motivic homotopy theory where varieties and CW-complexes are replaced by algebraic varieties and schemes. Such an extension has been actively developed,first as an essential piece of Voevodsky’s proof of the Milnor conjecture.

The talk will present these developments in stable motivic homotopy theory,from the successful extension of Quillen’s work by Levine and Morel,to emerging topics such as Panin and Walter’s orientation theories leading to quadratic enumerative geometry.

Lecture notes

Problem set

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The 2024 Program: Motivic Homotopy Theory
Organizers: Benjamin Antieau, Northwestern University; Marc Levine, Universität Duisberg-Essen; Oliver Röndigs, Universität Osnabrück; Alexander Vishik, University of Nottingham; and Kirsten Wickelgren, Duke University

Motivic homotopy theory arose out of the work of Morel and Voevodsky in the 1990s and since then has developed into both a powerful tool for understanding many arithmetic aspects in algebra and algebraic geometry, as well as being a fascinating generalisation of classical homotopy theory with an active development in its own right.

The 2024 GSS on motivic homotopy theory will give participants an introduction to some aspects of motivic homotopy theory as well as a taste of developments in other fields that have been influenced and enabled by motivic homotopy theory. Mini-courses will include: an introduction to unstable motivic homotopy theory, a study of characteristic classes in stable motivic homotopy theory, motivic homotopy theory in enumerative geometry, and a version of Weil conjectures in motivic homotopy theory,  as well as courses on recent advances in arithmetic properties of local systems, fundamental problems in Galois cohomology of fields, and aspects of G-bundles in algebraic geometry. 

Prerequisites: Students should have a basic knowledge of algebraic geometry,  algebraic topology, and some homotopy theory. For some of the courses, a knowledge of Galois cohomology and étale cohomology will also be helpful.   
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The Gradate Summer School at PCMI consists of a series of several interwoven minicourses on different aspects of the main research theme of that summer.  These courses are taught by leading experts in the field, chosen not only for their stature in the field but their pedagogical abilities. Each minicourse comprises three to five lectures. Each course is accompanied by a daily problem session, structured to help students develop facility with the material.
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The GSS takes place within the broader structure of PCMI, so there are many researchers at all levels in the field in attendance, as well as participants in the other PCMI programs.