Lie Theory without Groups:
Enumerative Geometry and Quantization of Symplectic Resolutions
Enumerative Geometry Beyond Numbers
Abstract for the Midrasha
Recently found answers to a number of questions in enumerative algebraic geometry are formulated in terms of highly nontrivial Lie theoretic structures. A prominent example is provided by calculation of quantum cohomology and quantum K-theory of some symplectic resolutions of singularities. On the other hand, quantizations of such resolutions form a natural generalization of enveloping algebras of semi-simple Lie algebras. Representation theory of these quantizations shows surprising connections to enumerative geometry of the resolution. It is also expected to be closely related to categorical invariants of the resolution studied in symplectic geometry, such as the Fukaya category. The goal of the workshop is to give an introduction to this circle of ideas and stimulate work towards conceptual understanding of the observed phenomena.
Abstract for the MSRI program
(Thanks to Roman Bezrukavnikov for telling me about it.)
Traditional enumerative geometry asks certain questions to which the expected answer is a number: for instance, the number of lines incident with two points in the plane (1, Euclid), or the number of twisted cubic curves on a quintic threefold (317 206 375). It has however been recognized for some time that the numerics is often just the tip of the iceberg: a deeper exploration reveals interesting geometric, topological, representation-, or knot-theoretic structures. This semester-long program will be devoted to these hidden structures behind enumerative invariants, concentrating on the core fields where these questions start: algebraic and symplectic geometry.