RUI: Analysis on HyperKahler Moduli Spaces

在几何学和物理学的许多领域，被研究对象的集合可以被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物体，从物理粒子到几何形状，再到微分方程的解。它们很重要，不仅因为它们携带了它们参数化的对象的信息，还因为它们代表了有趣的几何图形的自然发生的例子。基于四元数的代数，有几个众所周知的模空间族，它们带有所谓的超kahler几何，并且物理上的考虑表明，这些模空间可能在其整体形状或拓扑中反映出某些强性质。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，因为它接触了几个不同的分析领域，几何和拓扑，并为物理学中的某些对偶原理提供了一个试验场。该项目的其他影响包括促进PI的教育活动，包括监督原始的本科生研究，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的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，其中包括物理和几何中感兴趣的几个超kahler模空间族。磁单极子的模空间就是一个例子，它的L2上同调是来自超对称物理学的Ashoke Sen的一个长期存在且仍然开放的猜想的主题。这个开创性的猜想产生了几个关于相似模空间的平行猜想，并且是由物理学引起的许多有趣的几何问题的起点。本项目将开发的一个主要工具是将Mazzeo和Melrose在具有“纤维边界”的流形上著名的伪微分算子演算推广到具有“准纤维边界”的紧流形类别。这是单极模空间和其他超kahler模空间族允许自然紧化的范畴，其构造是本项目的第二个主要目标。从Sen猜想的证明开始，这项工作将使精细的几何微局部分析技术在超kahler几何及其他领域的这些重要空间类别的研究中获得新的可用性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.