CAREER: Geometric Applications of Gauge Theory

摘要奖：DMS-1151693首席研究员：Andrew Neitzke PI的研究重点是物理和几何之间的接口。在与Davide Gaiotto和Greg摩尔的长期合作中，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相当意外的物理和数学科目之间的联系，如枚举几何，hyperkahler几何和集群代数，因此也纯粹数学这些科目之间的联系。拟议的研究建立在最近的工作，并在几个方向上扩展。首先，在与Gaiotto和摩尔的持续合作中，PI将研究广义Donaldson-Thomas不变量的新例子，这些不变量“计数”黎曼曲面上的测地线轨迹的某些网络。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。此外，在合作与Gaiotto和摩尔，PI将继续发展一种新的方法，构建hyperkahler度量的总空间的可积系统。在这种新方法中，广义Donaldson-Thomas不变量起着关键作用，最终目标是产生渐近或收敛的K3度量的级数表示。最后，在独奏工作中，PI将探索新的数学结构，当N=2超对称场论研究时，将时空视为Taub-NUT空间。这一建设预计将导致新的见解科学的nonabelian θ函数。PI还将制作一个视频剪辑库，为各个层次的数学家（从本科生到其他研究人员）解释物理学概念。这些剪辑将在网上免费提供。此外，PI将继续他积极的教学计划，简要的写作和讲座，旨在更广泛地传播他的方法和结果。PI使用从粒子物理学导入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“场方程”，它控制着时空的曲率。这个由爱因斯坦首先写下的方程是出了名的难以处理。特别是，已知这个方程有一个描述封闭四维宇宙的大的解族（所谓的“K3度量”），但从来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，这将寻找K3度量（和其他类似解决方案）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发展。PI还将制作一个视频剪辑库，为各个层次的数学家（从本科生到其他研究人员）解释物理学概念。 这些剪辑将在网上免费提供。

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.