Interactions between representation theory, algebraic geometry, and physics

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Representation theory is an area of mathematics which deals with the study of symmetry, and is fundamental to many areas of scientific study. For example, the electron orbitals of a hydrogen atom are classified in part through the representation theory associated to the symmetries of a sphere. Representation theory also has diverse theoretical applications within mathematics itself. Some of the most modern developments in this field fall under the umbrella of higher representation theory. Here, problems are recast in geometric or abstract terms, allowing for the application of novel techniques. This often takes place in the framework of algebraic geometry: the study of solutions of systems of polynomial equations. The interplay between higher representation theory and algebraic geometry is extremely rich, and has led to the resolution of challenging problems such as the Kazhdan-Lusztig Conjectures. Less understood, but perhaps equally as rich, are emerging relations between higher representation theory and theoretical physics. Many of the spaces and concepts that are central to higher representation theory also appear in quantum field theory, and the perspective of physics opens the door to new proofs, techniques, and intuition. A prime example is Kapustin and Witten's physical interpretation of the Geometric Langlands program in terms of super Yang-Mills theory, which has inspired many mathematicians. The proposed research focuses on the interactions between higher representation theory, geometry, and physics. It will address natural questions which arise at the intersection of these fields, such as: - What problems and structures in higher representation theory can we understand using ideas from physics, and vice versa? - What are the special algebro-geometric properties of spaces arising in higher representation theory, and how do they relate to physical constructions? - How can we further expand the connections between higher representation theory and physics, and foster interdisciplinary collaboration? A primary focus of this proposal is Coulomb branches, which are spaces arising in quantum field theory. They were very recently given a rigourous mathematical definition by Braverman, Finkelberg, and Nakajima. Subsequent work has established ties between Coulomb branches and several areas of mathematics, including representation theory, algebraic geometry, number theory, quantum groups, and integrable systems. The proposed research aims to develop and understand these connections, and apply Coulomb branch techniques to resolve important mathematical problems in these fields. The results of this research will be of interest to mathematicians and to physicists, and contribute foundational results to a developing area of research. This research program offers opportunities for training at all levels. It will expose students and postdocs to a very active field of study, and foster the development of new research expertise in Saskatchewan.