Topics in Noncommutative Geometry

Prof. Yuri Berest (Cornell University)

October 11, 2012, 15.15 - 16.15, HG G 43
October 25, 2012, 15.15 - 16.15, HG G 43
November 1, 2012, 15.15 - 16.15, HG G 43
November 8, 2012, 15.15 - 16.15, HG G 43
November 9, 2012, 9.15 - 10.15, HG G 19.1

Abstract

In these lectures, we will discuss a variety of topics from algebraic geometry, noncommutative algebra and representation theory that usually go under the name of noncommutative geometry. Our aim is to give an overview of several popular approaches to noncommutative geometry and look at some interesting applications (mostly, in mathematical physics). Although the level of the lectures will be uneven (some topics will be covered in more detail while others will be touched upon only briefly), an effort will be made to make the discussion accessible for graduate students. A tentative syllabus:

Rings of Differential Operators on Algebraic Varieties
(Morita theory of differential operators. Differential operators on smooth and singular varieties. Differential equivalence and isomorphism. Ideals and commutative rings of differential operators on curves . Relation to integrable systems: Wilson's adelic Grassmannian and the KP hierarchy)

Noncommutative Projective Geometry, I. Introduction
(Coherent sheaves on projective varieties, Serre's Theorem. Artin-Zhang's noncommutative Proj. Twisted homogeneous coordinate rings. AS regular algebras associated to elliptic curves. Example: Sklyanin algebras. Classification of noncommutative projective planes and quadrics)

Noncommutative Projective Geometry, II. Applications
(Moduli spaces of framed torsion-free sheaves on NC projective planes and quiver varieties. The ADHM construction and the twistor transform in noncommutative geometry. Noncommutative instantons)

Smooth Algebras and the Representation Functor
(NC symplectic geometry. The formalism of double derivations: bisymplectic geometry and double Poisson brackets. Examples. Application: Calogero-Moser spaces over algebraic curves)

Homotopical Algebra and Noncommutative Geometry
(Quillen's model categories. Nonabelian derived functors. Pointed model categories: the loop and suspension functors. Examples: model categories of DG algebras and DG categories. Quillen cohomology. Applications: Ginzburg DG algebras, deformed Calabi-Yau completions (after Keller and Van den Bergh). Derived representation schemes)