Gauge theoretic moduli spaces

从数学的角度来看，规范理论是研究被称为矢丛的物体及其上的结构(例如，连接、希格斯场等)，这些结构满足物理学中的微分方程式。这种微分方程式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。夏博诺博士的研究计划主要是研究瞬子和单极子的模空间的几何和拓扑，并为他们的研究组装工具。 对于欧几里得空间上的单极子，无论有没有周期条件，一个称为Nahm变换的工具启发式地允许计算模空间。这种启发式方法已被证明在某些情况下有效，但夏博诺博士通过这笔赠款，旨在改善可以使用这种工具的条件。通过它，他打算解决长期存在的关于单极的模空间的猜想，其中一些来自物理，另一些来自数学。沙博诺博士早先曾成功地在其他环境下建立了纳姆变换，因此他很有能力解决这些问题。 目前感兴趣的一个主题是研究高维空间上的瞬子。这些空间以有趣的几何形状为特色，我们称之为“特殊完整”。唐纳森和托马斯提出了一个著名的研究计划，将这些空间上的规范理论与这些空间中特殊对象的子几何联系起来，称为定标子流形。目前，国际数学界对校准子几何方面的了解比对规范理论方面的了解更深，Charbonneau博士在这项拨款任期内的工作有助于促进对规范理论方面的了解。 夏博诺博士的研究项目的另一个方面是跨学科，他的几何专长有助于研究玻璃化转变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.