Mathematical physics

The mathematical definition of Coulomb branches of 3d N=4 gauge theories gives ring objects in the equivariant derived Satake category. We have another fundamental example of a ring object, namely the regular sheaf. It corresponds to the 3d N=4 QFT T[G], studied by Gaiotto-Witten. We also have operations on ring objects, corresponding to products, restrictions, Coulomb/Higgs gauging in the `category' of 3d N=4 QFT's. Thus we conjecture that arbitrary 3d N=4 QFT with G-symmetry gives rise a ring object in the derived Satake for G.

In this talk, I plan to review the global Langlands correspondence in the de Rham setting. The focus will be on the `big picture': the formulation of the correspondence, its expected properties, and possible approaches towards its proof.

An approximate representation of a finitely-presented group is an assignment of unitary matrices to the generators, such that the defining relations are close to the identity in the normalized Hilbert-Schmidt norm. A group is said to be hyperlinear if every non-trivial element can be bounded away from the identity in approximate representations of the group. Determining whether all groups are hyperlinear is a major open problem, as a non-hyperlinear group would provide a counterexample to the famous Connes embedding problem.

Given the difficulty of the Connes embedding problem, it makes sense to look at an easier problem: how fast does the dimension of approximate representations grow (as a function of how close the defining relations are to the identity) when we require a given set of elements to be bounded away from the identity. These growth rates are called the hyperlinear profile of the group.

In this talk, I will explain our best lower bounds on hyperlinear profile, as well as the connection to entanglement requirements for non-local games (joint work with Thomas Vidick). Time permitting,  I will also mention some other approaches to looking for non-hyperlinear groups, including the recent work of De Chiffre, Glebsky, Lubotzky, and Thom on a group which is not approximable in the unnormalized Hilbert-Schmidt norm.