Analysis, Geometry, and Mathematical Physics

涉及分析、几何和数学物理的跨学科研究对这些学科的最新进展都起到了重要作用，这些学科都取得了重大突破。分析是求解微分方程，特别是几何和物理中出现的非线性偏微分方程的主要工具。这些方程的解法反过来又为解答几何和物理中的深奥问题创造了新的工具。另一方面，几何学和物理学的深刻见解有助于分析的发展。这个研究项目调查了由弦理论驱动的三个领域的界面问题。预计该项目的结果将在几个方向上产生重要影响，包括代数几何和数论的进步，对镜像对称的加深理解，引力波的建模以及离散环境下微分几何的应用。本研究项目以Strominger、Yau和Zaslow开创的SYZ图为基础，探索Calabi-Yau流形之间的对偶理论。复几何与辛几何之间的关系对于理解全纯束和特殊拉格朗日子流形是非常重要的。弦理论背后的超对称几何是由非线性方程控制的。一些方程推广了厄米杨-米尔斯方程，尽管本项目研究的问题更非线性。目前的研究之一就是要确定这些微分方程的存在准则。另一个项目将探索辛上同调作为研究镜像对称的可能工具。一个正在进行的合作项目试图描述广义相对论中的准局部物理量。分析的思想也将应用于图论的研究。

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.