Gauge theoretic moduli spaces

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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.