Interdisciplinary research involving analysis, geometry, and mathematical physics has been instrumental for recent progress in each of these subjects, which have all witnessed major breakthroughs. Analysis is used as a major tool to solve differential equations, especially the nonlinear partial differential equations that arise in geometry and physics. Solution methods for those equations in turn create new tools for answering deep questions in geometry and physics. On the other hand, deep insights in geometry and physics help the development of analysis. This research project investigates questions at the interface of the three areas motivated by string theory. Results of the project are anticipated to have important consequences in several directions, including advances in algebraic geometry and number theory, enhanced understanding of mirror symmetry, modeling of gravitational waves, and application of differential geometry in discrete settings.
This research project explores the theory of duality between Calabi-Yau manifolds based on the SYZ picture pioneered by Strominger, Yau, and Zaslow. The relationship between complex geometry and symplectic geometry is very important to understanding of holomorphic bundles and special Lagrangian submanifolds. There are nonlinear equations governing the supersymmetric geometry behind string theory. Some of the equations generalize Hermitian Yang-Mills equations, although the questions under study in this project are more nonlinear. One of the current investigations is intended to determine existence criteria for these differential equations. Another project will explore symplectic cohomology as a possible tool to study mirror symmetry. An ongoing collaborative project seeks to describe quasilocal physical quantities in general relativity. Ideas of analysis will also be applied to study graph theory.
