RUI: Analysis on HyperKahler Moduli Spaces

在几何学和物理学的许多领域中，研究对象的集合可以组织在一起，形成一个具有内在几何的空间，称为“模空间”。模空间可以参数化各种各样的对象，从物理粒子到几何形状，再到微分方程的解。它们之所以重要，不仅是因为它们携带了关于它们所参数化的对象的信息，而且还因为它们代表了有趣的几何体的自然发生的示例。有几个著名的模空间家族，它们携带基于四元数代数的所谓超卡勒几何，物理考虑表明这些模空间可能具有某些强性质，反映在其全局形状或拓扑中。该项目的主要目标是开发分析工具来建立这些属性，并更深入地了解这些空间及其相关空间的几何形状。这项工作具有重要的跨学科吸引力，因为它与分析，几何和拓扑学的几个不同领域接触，并为物理学中的某些对偶原理提供了试验场。该项目的其他影响包括促进PI的教育活动，包括监督原始本科生研究，指导学生主导的基于项目的独立研究项目，以及正在进行的改善佛罗里达新学院数学课程多样性和性别平等的倡议。该项目将开发一个框架，用于分析一类空间上的椭圆算子，其中包括几个超Kahler模空间的物理和几何感兴趣的家庭。磁单极子的模空间就是一个例子，它的L2上同调是Ashoke Sen从超对称物理学中提出的一个长期存在且仍然开放的猜想的主题。这个开创性的猜想催生了几个关于相似模空间的平行命题，并且是许多有趣的物理几何问题的起点。该项目将开发的一个主要工具是Mazzeo和Melrose的著名微积分的伪微分算子流形上的“纤维边界”，一类紧凑的流形与“准纤维边界”的推广。这是一个范畴，在这个范畴中，超Kahler模空间和其他超Kahler模空间家族允许自然紧化，其构造是这个项目的第二个主要目标。从森猜想的证明开始，这项工作将因此使精细的几何微局部分析技术新提供给这些重要的空间类的研究hyperKahler几何和beyond.This奖反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

In many areas of geometry and physics, the collection of objects under investigation can be organized together to form a space with intrinsic geometry, known as a "moduli space". Moduli spaces may parameterize a wide variety of objects, from physical particles, to geometric shapes, to solutions of differential equations. They are important not only because they carry information about the objects they parameterize, but also because they represent naturally occurring examples of interesting geometries. There are several well-known families of moduli spaces which carry a so-called hyperKahler geometry, based on the algebra of quaternions, and physical considerations suggest that these moduli spaces may have certain strong properties reflected in their global shape, or topology. A principal goal of this project is to develop analytical tools to establish these properties and more deeply understand the geometry of these spaces and their relatives. This work has significant interdisciplinary appeal as it makes contact with several different areas of analysis, geometry and topology, and provides a testing ground for certain duality principles in physics. Additional impacts of the project include the facilitation of the PI's educational activities, including the supervising of original undergraduate research, mentoring student-led project-based independent study projects, and an ongoing initiative to improve diversity and gender parity in the mathematics program at New College of Florida.This project will develop a framework for the analysis of elliptic operators on a class of spaces which includes several families of hyperKahler moduli spaces of interest in physics and geometry. These are exemplified by the moduli spaces of magnetic monopoles, the L2 cohomology of which is the subject of a long standing and still open conjecture of Ashoke Sen coming from supersymmetric physics. This seminal conjecture spawned several parallel conjectures about similar moduli spaces, and is the starting point for a number interesting geometric questions motivated by physics. A principal tool this project will develop is a generalization of Mazzeo and Melrose's well-known calculus of pseudodifferential operators on manifolds with "fibered boundary", to a category of compact manifolds with "quasi-fibered boundary". This is a category in which the monopole moduli spaces and other families of hyperKahler moduli spaces admit natural compactifications, the construction of which is a second major goal of this project. Beginning with a proof of Sen's conjecture, this work will therefore make refined techniques of geometric microlocal analysis newly available to the study of these important classes of spaces in hyperKahler geometry and beyond.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.