Contraction algebras and noncommutative derived geometry

Matt Booth (University of Edinburgh)

Wednesday 20th February 16:00-17:00 Maths 311B

Abstract

Given a threefold flopping contraction, one can associate to it a finite-dimensional noncommutative algebra, the contraction algebra, which controls the noncommutative deformation theory of the flopping curves. If the threefold was smooth, this algebra is conjectured to determine the complete local geometry of the base. I'll talk about a new invariant, the derived contraction algebra (which has an interpretation in terms of derived deformation theory), and explain (via singularity categories) why the derived version of the above conjecture holds. Time permitting, I'll talk about the flop-flop autoequivalence and indicate some aspects of the theory for surfaces.

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