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. 051119 Abstract (This will be a nontechnical 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 Voevodskyâ€™s proof of the Milnor and BlochKato conjectures. In this minicourse, 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 nontechnical 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.
