I will speculate about connections between partition functions in lattice gauge theory, the approach to representation theory of the symmetric group due to Okounkov-Vershik, and geometry of surfaces a la Magee-Puder (in progress, joint with V. Chelnokov).
An LMS online lecture course in gauge theory
An LMS online lecture course in branes, gauge theories and dualities.
In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.
In the 1970s, Galewski-Stern and Matumoto studied the existence and the classification of triangulations on topological manifolds of dimension at least 5. They reduced these problems to questions about the three-dimensional homology cobordism group, ΘH3, and the Rokhlin homomorphism from this group to ℤ/2. The structure of the homology cobordism group is still unknown, but some information can be obtained using tools from gauge theory and symplectic geometry, such as the Seiberg-Witten Floer spectrum and involutive Heegaard Floer homology. I will describe the proof of the existence of non-triangulable high-dimensional manifolds (using gauge theory), and some open problems.
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