Gauge theoretic moduli spaces

从数学的角度来看，规范理论是研究被称为向量束的物体及其上的结构（例如，连接，希格斯场等）满足从物理学推导出的微分方程。这样的微分方程的许多解可以被认为是等价的，数学家和物理学家把不等价解的参数空间称为模空间。Charbonneau博士的研究项目主要是研究瞬子和单极子的模空间的几何和拓扑结构，以及为他们的研究组装工具。对于欧几里得空间上的单极子，不管是否有周期条件，一种叫做纳姆变换的启发式工具允许对模空间进行计算。这种启发式方法已被证明在某些情况下是有效的，但Charbonneau博士通过这笔资助，旨在改善这种工具的使用条件。通过它，他打算解决长期以来关于单极子模空间的猜想，其中一些来自物理学，另一些来自数学。由于他早期在其他环境中成功地建立了纳姆变换，夏博诺博士在解决这些案件方面已经很有经验了。当前人们感兴趣的一个主题是高维空间上的瞬子的研究。这些空间具有有趣的几何形状，我们称之为“特殊完整”。Donaldson和Thomas提出了一个著名的研究项目，将这些空间的规范理论与这些空间中特殊物体的子几何联系起来，称为校准子流形。目前，国际数学界对校准子几何方面的理解比规范理论方面的理解更深，而Charbonneau博士在这项资助的任期内的工作有助于推进规范理论方面的知识。Charbonneau博士的研究项目的另一个方面是跨学科的，他的几何专业知识促进了研究玻璃跃迁的软物质科学家的研究。

From a mathematical perspective, gauge theory is the study of objects called vector bundles and structures on them (e.g., connections, Higgs fields, etc) satisfying differential equations derived from physics. Many solutions for such differential equations may be considered equivalent, and mathematicians and physicists call the parameter space of inequivalent solutions the moduli space. Dr. Charbonneau's research program is mostly concerned with studying the geometry and topology of moduli spaces of instantons and monopoles, and in assembling tools for their study. For monopoles on Euclidean spaces, with periodic conditions or not, a tool called the Nahm transform heuristically allows for a computation of the moduli space. This heuristic has been proven to work in some settings, but Dr. Charbonneau, through this grant, aims to improve the conditions under which this tool can be used. Through it, he intends to settle longstanding conjectures about the moduli spaces of monopoles, some of them coming from physics, others from mathematics. With his earlier successes in establishing the Nahm transform in other settings, Dr. Charbonneau is well established to settle those cases. A current theme of interest is the study of instantons on spaces of high dimensions. These spaces feature interesting geometries we call "special holonomy." There is a celebrated research program proposed by Donaldson and Thomas relating the gauge theory on those spaces to the sub-geometry of special objects in these spaces called calibrated sub-manifolds. At the moment, the international mathematical community has a deeper understanding on the calibrated sub-geometry side than on the gauge theory side and Dr. Charbonneau's work during the tenure on this grant helps advance knowledge on the gauge theory side. Another aspect of Dr. Charbonneau's research program is interdisciplinary, where his geometric expertise facilitates the research of soft-matter scientist who study glass transition.