CAREER: Differential Equations, Algebraic Geometry, and String Theory

宇宙的结构是由数学编码的。 例如，宇宙的大尺度行为由爱因斯坦的广义相对论决定，而小尺度行为则由量子力学决定。 理论物理学的一个突出目标是找到一个统一这些大尺度和小尺度描述的宇宙的数学描述。 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。 弦理论的方程既困难又复杂，人们期望它能将数学和物理学的不同分支联系起来。理解这些方程及其所包含的结构是这些教育和研究项目的主要目标。 这个研究议程旨在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。 计划的活动包括赞助本科生的研究项目，一年两次的研究生周末会议，以及为期两周的暑期项目，包括为期一周的暑期学校，研究生和高年级本科生与现有的暑期研究项目垂直整合。这些项目旨在研究几个几何问题，这些问题受到弦理论的启发，并涉及微分方程，微分几何和代数几何。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。 我们的目标是开发一个理论，涉及到这些方程的研究，在代数几何结构，使用无限维版本的经典几何不变理论。 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，以了解在某些奇异的几何对象编码的代数结构，在类比与乘子理想和奇异的，正弯曲的度量全纯线丛之间的连接。 因此，这个项目将涉及发展偏微分方程，以及微分和代数几何的新思想。 这些想法将在各种简化模型中进行测试，这将导致解决卡勒几何中的几个开放问题。 另一项工作是研究弦理论中的突现引力现象，特别是AdS/CFT猜想，以及AdS/CFT的组合方面。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.