MOTIVIC HOMOTOPY THEORY and APPLICATIONS
3.30 pm, Lecture Hall 5
Lecture 1 :Some contributions of motivic homotopy theory to algebraic geometry
Anand Sawant
TIFR.
05-11-19

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Abstract

(This will be a non-technical introduction and a brief survey of applications.)

Since its inception in the foundational work of Morel and Voevodsky in the 1990's, motivic (or A^1)-homotopy theory has provided a systematic framework to successfully adapt several techniques of algebraic topology to the realm of algebraic geometry by having the affine line A^1 play the role of the unit interval. In the last 20 years, motivic homotopy theory has led to spectacular applications such as Voevodsky’s proof of the Milnor and Bloch-Kato conjectures. In this mini-course, I will introduce motivic homotopy theory and discuss some of its applications with emphasis on algebraic groups.

Title: Some contributions of motivic homotopy theory to algebraic geometry.

Abstract: I will discuss a few homotopy invariance phenomena in commutative algebra and algebraic geometry and explain how they can be approached using a suitable homotopy theory for algebraic varieties. The emphasis will be on two questions about algebraic groups, which have their origins in classical questions on projective modules. I will also give a non-technical introduction to motivic homotopy theory developed by Morel and Voevodsky. In the subsequent lectures, I will formally introduce motivic homotopy theory and describe some of the ideas that go into the proofs of these results.