**Canonical bases and algebraic geometry**

Roman Bezrukavnikov (MIT)

**Abstract:**

Connections between algebra and geometry are good not just because they help us become more complete human beings by unifying our left-hemispheric and right-hemispheric perception of mathematics, but also because they allow to prove new results in one of the areas by using the methods of the other. Representation theory, a branch of algebra, seeks to understand modules over specific rings, such as the universal enveloping of a Lie algebra. Geometric representation theorists accomplish this by realizing the module as a sort of integral of a family of modules (more precisely, global sections of a sheaf of modules, or a complex of such) parametrized by points of an algebraic variety. Known features of geometry of the variety then provide rather subtle algebraic information about the modules. I will describe some old (Kazhdan-Lusztig theory) and new (representations of quantized symplectic resolutions) examples of this situation. Time permitting I will discuss attempts to reverse the direction of the flow of applications and produce results in algebraic geometry inspired by representation theory.

Wednesday, Nov. 5, 4:30 – 5:30 p.m.

Mathematics 520

Tea will be served at 4:00 p.m.