CAREER: Geometric Applications of Gauge Theory

项目编号：dms -1151693项目负责人：Andrew Neitzke PI的主要研究方向是物理与几何之间的界面。在与Davide Gaiotto和Greg Moore的长期合作中，他研究了N=2超对称量子场论的数学应用。这项工作最近导致了物理学和数学学科之间意想不到的联系，如枚举几何、超kahler几何和聚类代数，因此也导致了这些学科之间的纯数学联系。拟议的研究在几个方向上建立并扩展了最近的工作。首先，在继续与Gaiotto和Moore的合作中，PI将研究广义Donaldson-Thomas不变量的新例子，它“计数”黎曼曲面上的某些测地线轨迹网络。这个计数问题对应于确定某些物理理论中稳定粒子的数量。PI还将与Gaiotto和Moore合作，继续开发一种在可积系统的总空间上构造超kahler度量的新方法。在这种新方法中，广义Donaldson-Thomas不变量起着关键作用；最终目标是产生K3度量的渐近或收敛级数表示。最后，在个人工作中，PI将探索在研究以时空为Taub-NUT空间的N=2超对称场论时出现的新的数学结构。这种构造有望导致对非abel theta函数科学的新见解。PI还将制作一个视频剪辑库，为不同水平的数学家（从本科水平到其他研究人员）解释物理概念。这些片段将在网上免费提供。此外，PI将继续他积极的教学计划，说明文写作和演讲，旨在更广泛地传播他的方法和成果。PI使用从粒子物理学引进的工具研究几何问题。最近，他和他的合作者设计了一个新的方案来解决控制时空曲率的“场方程”。这个由爱因斯坦首先写下的方程，是出了名的难解。特别是，众所周知，这个方程有一个描述封闭四维宇宙的大解族（所谓的“K3度量”），但没有人能够写出代表这些解的实际公式。PI和合作者发现了爱因斯坦方程和粒子物理学之间令人惊讶的联系，它将寻找K3度量（以及其他类似的解决方案）的问题与理解亚原子粒子如何衰变的问题联系起来。这种联系导致了对这两个问题的深入了解，PI和合作者现在正在积极开发这些问题。PI还将制作一个视频剪辑库，为不同水平的数学家（从本科水平到其他研究人员）解释物理概念。这些片段将在网上免费提供。

AbstractAward: DMS-1151693Principal Investigator: Andrew Neitzke The PI's research is focused on the interface between physics and geometry. In a long-running collaboration with Davide Gaiotto and Greg Moore, he has studied mathematical applications of N=2 supersymmetric quantum field theory. This work has recently led to quite unexpected connections between physics and mathematical subjects such as enumerative geometry, hyperkahler geometry, and cluster algebras, and hence also to purely mathematical connections between these subjects. The proposed research builds on and extends this recent work in several directions. First, in continuing collaboration with Gaiotto and Moore, the PI will investigate new examples of generalized Donaldson-Thomas invariants, which "count" certain networks of geodesic trajectories on Riemann surfaces. This counting problem corresponds to determining the numbers of stable particles in certain physical theories. Also in collaboration with Gaiotto and Moore, the PI will continue development of a new method for constructing hyperkahler metrics on total spaces of integrable systems. In this new method the generalized Donaldson-Thomas invariants play a key role; the eventual goal is to produce asymptotic or convergent series representations for K3 metrics. Finally, in solo work, the PI will explore new mathematical structures which appear when the N=2 supersymmetric field theory is studied taking spacetime to be Taub-NUT space. This construction is expected to lead to new insights into the science of nonabelian theta functions. The PI will also produce a library of video clips, explaining concepts from physics for mathematicians at a variety of levels (from undergraduate level to other researchers). These clips will be freely available over the Web. In addition the PI will continue his active program of teaching, expository writing and talks, aimed at disseminating his methods and results more broadly.The PI studies geometric problems using tools imported from particle physics. Recently, he and his collaborators have devised a new scheme for solving the "field equation" which governs the curvature of spacetime. This equation, first written down by Einstein, is notoriously intractable. In particular, it is known that this equation has a large family of solutions describing a closed four-dimensional universe (so-called "K3 metrics"), but nobody has ever been able to write down actual formulas representing these solutions. The PI and collaborators have found a surprising connection between Einstein's equation and particle physics, which relates the problem of finding K3 metrics (and other similar solutions) to the problem of understanding how subatomic particles can decay. This connection has led to insight into both problems, which the PI and collaborators are now actively developing. The PI will also produce a library of video clips, explaining concepts from physics for mathematicians at a variety of levels (from undergraduate level to other researchers). These clips will be freely available over the Web.