CAREER: Finiteness for Hyperkahler Manifolds

超凯勒流形是基于四元数的特殊对称性的几何空间，四元数是汉密尔顿在1843年发现的复数的四维模拟。超凯勒流形在高能物理、量子场论和弦理论中的普遍性是非常显著的。它们作为杨-米尔斯瞬子、磁单极子、希格斯束和许多其他物理方程的解的参数空间自然出现。私家侦探将研究几何和拓扑的hyperkähler流形。他将构建新的例子hyperkähler流形，和/或他们的奇异对应物，hyperkähler orbifolds。他还将建立新的限制可能的拓扑类型的hyperkähler流形。该项目将有助于更好地理解一类几何空间，这些空间是许多物理模型的核心，也连接了不同的数学领域，包括代数和微分几何，拓扑学和数论。私家侦探将指导博士，硕士和本科荣誉学生，谁将协助研究项目。他将通过组织高级专题小型学校、促进学生主导的研讨会以及使几何学和拓扑学课程现代化来增加北卡罗来纳州大学研究生的培训机会。在本科阶段，他将领导一个解决问题的研讨会，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发起一个新的海外留学暑期项目。他将通过积极招募第一代大学生和其他代表性不足的群体的学生来倡导多样性，参与这些非传统的活动。超凯勒流形的结构及其在物理学中的应用已经得到了很好的研究，但只有少数紧凑的例子是已知的：在每个维度上只有两个或三个变形类。同时，不知道在每个维度中可能有多少变形类。私家侦探是由证明这个数是有限的问题激发的。他的目的是表明，每一个hyperkähler流形可以变形为拉格朗日纤维化，hyperkähler流形承认全纯纤维空间结构。然后，他计划建立一般有限的结果，完善他早期的结果拉格朗日纤维。他将利用紧凑和非紧凑拉格朗日纤维化之间的类比，如希钦系统，以找到新的例子。私家侦探还将通过探索上同调环的结构来展示超凯勒流形上的一般拓扑边界。最终目标是更完整地理解超凯勒流形的可能拓扑。

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.