NSF Award
2305257
This project is concerned with a class of mathematical objects called symplectic resolutions. Symplectic resolutions are mathematical jewels; they possess a wealth of interesting structures, which are of interest to mathematicians and theoretical physicists working in different areas. As a result, new discoveries about symplectic resolutions often have a wide impact across many parts of mathematics and physics. As a mathematician working in an area called symplectic geometry, the PI (along with collaborators) will study certain algebraic structures associated to symplectic resolutions called Fukaya categories. The PI also will mentor undergraduate and graduate students wishing to enter this area of mathematics. To this end, the PI will write publicly available lecture notes aimed at advanced undergraduate or beginning graduate students.<br/><br/>Fukaya categories of symplectic manifolds are a class of algebraic structures defined via nonlinear analysis, by counting pseudoholomorphic curves with appropriate Lagrangian boundary conditions. There are long-standing expectations that Fukaya categories of symplectic resolutions should be related to structures arising in representation theory. The PI intends to work on several questions in this direction, using techniques from microlocal sheaf theory. Such techniques have only recently become available thanks to the fundamental work of Ganatra--Pardon--Shende, and the PI believes that they are particularly promising for these types of questions. Along the way, the PI also plans to further develop the foundations of microlocal sheaf theory, in particular the theory of perverse microlocal sheaves.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
