Chris Elliott – Research Statement
نویسنده
چکیده
One of the most central problems for 21st century mathematical physics is the question: “how can we mathematically formalize the notion of a non-perturbative quantum field theory?” Formulations of aspects of quantum field theory developed by mathematicians (e.g. [Seg88], [Ati88], [FLM89], [Lur09], [BD04]) have generated conceptual frameworks for problems in mathematics far removed from physics. What’s more, in recent decades the number of potential purely mathematical applications of quantum field theory has grown dramatically, with many striking claims made by physicists that dualities between quantum field theories have, as consequences, deep equivalences between algebro-geometric, representation theoretic and topological objects (see e.g. [Kon94], [SYZ96], [KW06], [AGT10], [MO12], [BFN14], [MNOP06a], [MNOP06b], [GV98]). As such, a deeper formal mathematical understanding of physically interesting quantum field theories and the dualities between them has the potential to lead to progress on fundamental problems in these fields. My work addresses aspects of this formalization problem in several critical examples, principally the example of N = 4 supersymmetric gauge theory.
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