CAREER: Differential Equations, Algebraic Geometry, and String Theory

宇宙的结构是由数学编码的。例如，宇宙的大尺度行为是由爱因斯坦的广义相对论决定的，而小尺度行为是由量子力学决定的。理论物理学的一个突出目标是找到一种将大尺度和小尺度描述统一起来的宇宙的数学描述。这项工作导致了弦理论的发现，以及大量美丽而复杂的数学结构和方程。弦理论的方程既难又复杂，有望将数学和物理的不同分支联系起来。理解这些方程式及其包含的结构，是这些教育和研究项目的主要目标。这项研究的目的是理解弦理论中物理现象的数学描述，从粒子的形成和衰变到“全息”弦理论中引力的出现。教育计划旨在为本科生和研究生提供参与与研究计划相关的学习和研究项目的机会。计划的活动包括为本科生赞助的研究项目，为研究生举办的一年两次的周末会议，以及一个为期两周的暑期项目，包括为期一周的暑期班和研究生以及与现有暑期研究项目垂直结合的高级本科生。这些项目旨在研究几何中的几个问题，这些问题受到弦理论的启发，涉及微分方程式、微分几何和代数几何的技术。第一个项目涉及研究镜像对称中产生的两个方程：变形的厄米-杨-米尔斯方程和特殊的拉格朗日方程。其目标是发展一种理论，将这些方程的研究与代数几何中的结构联系起来，使用经典几何不变量理论的无限维版本。这些方程提出了令人兴奋的新的数学挑战，从完全非线性系统的分析到理解某些奇异几何对象中编码的代数结构，类似于全纯线丛上乘子理想与奇异、正弯曲度量之间的联系。因此，这个项目将涉及发展偏微分方程式以及微分和代数几何的新想法。这些想法将在各种简化的模型中进行测试，这将导致卡勒几何中的几个公开问题的解决。另一项工作将研究弦理论中的浮现引力现象，特别是ADS/CFT猜想，以及与ADS/CFT的组合方面的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The structure of the universe is encoded by mathematics. For example, the large scale behavior of the universe is determined by Einstein's theory of general relativity, while the small scale behavior is dictated by quantum mechanics. A prominent goal of theoretical physics has been to find a mathematical description of the universe which unites these large and small scale descriptions. This work has led to the discovery of string theory, and a vast array of beautiful and intricate mathematical structures and equations. The equations of string theory are difficult and complicated, and are expected to connect radically different branches of mathematics and physics. Understanding these equations, and the structure they contain, is the primary goal of these education and research projects. This research agenda aims to understand mathematical descriptions of physical phenomena in string theories ranging from particle formation and decay to the emergence of gravity in "holographic" string theories. The education plans aim to provide opportunities for undergraduate and graduate students to engage in learning and research projects related to the research program. Planned activities include sponsored research projects for undergraduates, a bi-annual weekend conference for graduate students, and a two week summer program, including a week long summer school and graduate students and advanced undergraduates vertically integrated with existing summer research programs.These projects aim to investigate several problems in geometry which are inspired by string theory, and which involve techniques from differential equations, differential geometry, and algebraic geometry. The first project involves studying two equations arising in mirror symmetry: the deformed Hermitian-Yang-Mills equation, and the special Lagrangian equation. The goal is to develop a theory that relates the study of these equations to structures in algebraic geometry, using an infinite dimensional version of classical geometric invariant theory. These equations present exciting new mathematical challenges ranging from the analysis of fully nonlinear systems to understanding the algebraic structure encoded in certain singular geometric objects, in analogy with connections between multiplier ideals and singular, positively curved metric on holomorphic line bundles. As a result, this project will involve developing new ideas in partial differential equations, as well as differential and algebraic geometry. These ideas are to be tested in a variety of simplified models, which will lead to the resolution of several open problems in Kahler geometry. Another line of work will study the phenomenon of emergent gravity in string theory, and in particular the AdS/CFT conjecture, and connection with combinatorial facets of AdS/CFT.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.