Mathematical Physics

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.

Recently, Roland van der Veen and myself found that there are sequences of solvable Lie algebras "converging" to any given semi-simple Lie algebra (such as sl(2) or sl(3) or E8). Certain computations are much easier in solvable Lie algebras; in particular, using solvable approximations we can compute in polynomial time certain projections (originally discussed by Rozansky) of the knot invariants arising from the Chern-Simons-Witten topological quantum field theory. This provides us with the first strong knot invariants that are computable for truly large knots.  But sl(2) and sl(3) and similar algebras occur in physics (and in mathematics) in many other places, beyond the Chern-Simons-Witten theory. Do solvable approximations have further applications?

This is a repeat of a talk I gave in McGill University in February, 2017. A video recording, a handout, and some further links are at McGill-1702

I will give a brief survey of the study of decomposable Specht modules for the symmetric group and its Hecke algebra, which includes results of Murphy, Dodge and Fayers, and myself. I will then report on an ongoing project with Louise Sutton, in which we are studying decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’.

Let be a complex semisimple algebraic group of adjoint type and the wonderful compacti
cation. We show that the closure in \overline{G} of the centralizer of a regular nilpotent is isomorphic to the Peterson variety. We generalize this result to show that for any regular , the closure of the centralizer in is isomorphic to the closure of a general -orbit in the flag variety. We consider the family of all such centralizer closures, which is a partial compactication of the universal centralizer. We show that it has a natural log-symplectic Poisson structure that extends the usual symplectic structure on the universal centralizer.

In this talk, I'd like to explain how quantum integrability and (q-deformed) W-algebraic structure arise from the moduli space of quiver gauge theory. It'll be also shown that our construction gives rise to a new family of W-algebras.