Abstract

Symplectic representation theory -- the study of representation theory via symplectic algebraic geometry — has undergone staggering progress in the past two years. This includes groundbreaking work by Braverman, Finkelberg, and Nakajima on the definition and properties of the Coulomb branch, certain moduli spaces appearing in three-dimensional gauge theory and whose duality with Higgs branch, which are generalizations of Nakajima quiver varieties, extends in certain cases the mathematical symplectic duality proposed by Braden, Licata, Proudfoot, and Webster, and others. It also includes new conjectures and proofs of conjectures by Bezrukavnikov, Losev, and others on quiver varieties, their representation theory and ‘wall-crossing’ phenomena, and their relationship to symplectic duality, as well as progress by Maulik, Okounkov, and others on quantum cohomology of the same varieties, and further progress on generalizations of Beilinson and Bernstein’s localization theorem in the context of symplectic resolutions of singularities. This conference intends to bring together experts on all aspects of the theory. Speakers will be invited to explain the latest developements regarding: - The definition, and fundamental properties, of the Coulomb branch, its quantizations, and three-dimensional mirror symmetry - Breakthroughs in our understanding of quantum cohomology, and its K-theoretic counterpart - (Geometric) categorification, and applications to representation theory (in particular, combinatorial representation theory) - The role of gauge theory, topological field theories, and categories of branes in symplectic representation theory - The classification and structure of symplectic singularities, their symplectic resolutions, and quantizations ​These topics cover the entire breadth of symplectic representation theory, and will present participants with a broad overview of the subject, crystallizing the deep connections between each of its many strands.