CAREER: Finiteness for Hyperkahler Manifolds

Hyperkähler流形是具有基于四元数的特殊对称性的几何空间，四元数是哈密尔顿在1843年发现的复数的四维模拟。Hyperkähler流形因其在高能物理、量子场论和弦理论中的普遍存在而引人注目。它们自然地作为杨-米尔斯瞬子、磁单极子、希格斯丛和许多其他物理方程的解的参数空间而出现。P.I.将研究Hyperkähler流形的几何和拓扑。他将构造超kähler流形和/或其奇异对应的新例子，即超kähler或双流形。他还将对Hyperkähler流形可能的拓扑类型建立新的限制。这个项目将有助于更好地理解一类几何空间，这些空间位于许多物理模型的核心，也连接了数学的不同领域，包括代数和微分几何、拓扑学和数论。P.I.将指导博士、硕士和本科生荣誉学生，他们将协助研究项目。他将加强北卡罗来纳大学研究生的培训机会，组织关于高级主题的迷你学校，促进学生主导的研讨会，并使几何和拓扑学课程现代化。在本科阶段，他将领导一个问题解决研讨会，指导学生参加数学竞赛，通过荣誉项目促进研究，并为数学专业和潜在的数学专业学生启动一个新的海外学习暑期计划。他将通过积极招募第一代大学生和其他代表不足的群体的学生来参与这些非传统活动来倡导多样性。人们对超卡勒流形的结构及其在物理中的应用进行了很好的研究，但很少有紧凑的例子：每个维度只有两到三个形变类。同时，还不知道每个维度中可能有多少个变形类别。私家侦探的动机是证明这个数字是有限的。他的目的是证明每一个超kähler流形都可以变形为拉格朗日纤维，一个允许全纯纤维空间结构的超kähler流形。然后，他计划通过改进他关于拉格朗日函数的早期结果来建立一般的有限性结果。他将利用紧致拉格朗日纤颤和非紧致拉格朗日纤颤之间的类比，例如希钦系统，寻找新的例子。P.I.还将通过探索上同调环的结构来证明Hyperkähler流形上的一般拓扑界。最终目标是更完整地理解Hyperkähler流形的可能拓扑。

Hyperkähler manifolds are geometric spaces with special symmetries based on the quaternions, the four-dimensional analogue of the complex numbers discovered by Hamilton in 1843. Hyperkähler manifolds are remarkable for their ubiquity in high energy physics, quantum field theory, and string theory. They arise naturally as parameter spaces for Yang-Mills instantons, magnetic monopoles, Higgs bundles, and solutions of many other physical equations. The P.I. will study the geometry and topology of hyperkähler manifolds. He will construct new examples of hyperkähler manifolds, and/or their singular counterparts, hyperkähler orbifolds. He will also establish new restrictions on the possible topological types of hyperkähler manifolds. This project will contribute to a better understanding of a class of geometric spaces that lie at the heart of many physical models, and which also connect different areas of mathematics including algebraic and differential geometry, topology, and number theory. The P.I. will mentor PhD, masters, and undergraduate honors students, who will assist with the research project. He will enhance the training opportunities available to graduate students at the University of North Carolina by organizing mini-schools on advanced topics, by promoting student-led seminars, and by modernizing the geometry and topology courses. At the undergraduate level he will lead a problem solving seminar to coach students for mathematics competitions, facilitate research through honors projects, and initiate a new study abroad summer program for math majors and potential math majors. He will advocate for diversity by actively recruiting first generation college students and students from other under-represented groups to participate in these non-traditional activities.The structure of hyperkähler manifolds, and their applications in physics, are well studied, yet only few compact examples are known: just two or three deformation classes in each dimension. At the same time, it is not known how many deformation classes there might be in each dimension. The P.I. is motivated by the problem of showing that this number is finite. He aims to show that every hyperkähler manifold can be deformed to a Lagrangian fibration, a hyperkähler manifold admitting a holomorphic fibre space structure. He then plans to establish general finiteness results by refining his earlier results for Lagrangian fibrations. He will exploit the analogies between compact and non-compact Lagrangian fibrations, such as Hitchin systems, to find new examples. The P.I. will also demonstrate general topological bounds on hyperkähler manifolds by exploring the structure of the cohomology ring. The ultimate goal is a more complete understanding of the possible topologies of hyperkähler manifolds.