CAREER: Aspects of Microlocal Geometry

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In modern mathematics, symplectic geometry is the abstract setting for classical mechanics, and microlocal analysis is a tool used to study solutions to systems of differential equations. Microlocal sheaf theory is the geometric formulation of microlocal analysis. Recent advances have uncovered striking applications of microlocal sheaf theory to symplectic geometry. This project will explore and extend these applications in various directions, including further developing algebraic and combinatorial connections between curves, knots, and surfaces. In terms of education, the principal investigator will prepare much-needed elementary expository works on the subject of microlocal sheaf theory.The specific research goals of the proposal are twofold. First, to localize the calculations of the Fukaya category of a Weinstein manifold (a certain sort of exact symplectic manifold) to its Lagrangian skeleton. Once localized, performing this calculation is a matter of combinatorics; thus secondly, the PI intends to explore the resulting combinatorics. It is already known that the sort of combinatorics that arises has a representation-theoretic flavor; in particular cluster algebras and categorified knot invariants appear. Further insights in these subjects will be gained by exploring their connection with symplectic geometry in terms of the above-mentioned skeleta. The localization and calculation of these Fukaya categories that are the focus of this research will play an essential role in an approach to the homological mirror symmetry conjecture.