Representation theoretic methods in geometry and mathematical physics

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I have recently been exploring the category of coherent sheaves on the affine Grassmannian. It is natural to call this the coherent Satake category because its constructible analogue (the category of constructible sheaves on the affine Grassmannian) is usually called the Satake category.******The usual (constructible) Satake category is a very rich object in both number theory and geometric representation theory. In particular, it is closely related to the Langlands program and, by relatively recent results of Kapustin and Witten, to certain gauge theories in mathematical physics. In this interpretation, the famous Langlands duality phenomenon corresponds to electric-magnetic duality.******The coherent Satake category is likewise related to a gauge theory. However, this theory behaves differently and the mathematical counterpart of this story is relatively poorly understood. For example, the constructible Satake category is semisimple and its monoidal structure symmetric. In contrast, the coherent Satake category is neither semisimple nor symmetric. Instead, its structure can be described (conjecturally) as a monoidal cluster category.******In a recent preprint with H. Williams, we define and study this cluster structure for the affine Grassmannian of GL(n). The K-theory of this affine Grassmannian is the simplest example of a Coulomb branch of a 4d N=2 gauge field theory. The appearance of such cluster structures has been noticed more generally for other gauge field theories. Our proof relies heavily on the construction of a renormalized r-matrix which makes sense in any monoidal category whose product is compatible with an auxiliary chiral category. This suggests a to make progress in the study and understanding of other Coulomb branches of such field theories.******One of the main aims of this proposal is to develop new tools (geometric as well as representation-theoretic) in order to extend our results. For instance, an obvious goal is to define a cluster structure for affine Grassmannians of other groups. The coherent Satake category is also closely related to the convolution spaces studied earlier with J. Kamnitzer in the context of defining homological knot invariants (e.g. Khovanov homology) and a quantum K-theoretic version of geometric Satake. In those instances various tools from geometry and representation theory were developed and applied. One tool that stands out is the idea of categorical actions of quantum groups. We plan to develop further and adapt some of these techniques to the context of the coherent Satake category (and other categories arising from gauge field theories).*****