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November 5: Roman Bezrukavnikov (MIT)

Canonical bases and algebraic geometry

Roman Bezrukavnikov (MIT)

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.

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