The Yang-Mills model is a theoretical framework for fundamental forces and elementary particles. It has made deep impacts in various branches of mathematics. A key challenge in mathematical physics is to construct the quantum Yang-Mills theory on four dimensional space and prove the existence of a "mass gap". In this talk, we will discuss stochastic quantization i.e. Langevin dynamics of the Yang-Mills model on two and three dimensional tori. This is a stochastic process on the space of "gauge orbits", induced by the solution to a nonlinear Lie algebra-valued stochastic PDE driven by space-time white noise. The presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by “solutions”, “state space”, “orbit space” and “gauge invariant observables”. We rigorously construct these objects by combining techniques from analysis, PDE, Stochastic PDE,  and especially the theory of regularity structures. The talk is based on joint work with Ajay Chandra, Ilya Chevyrev, Martin Hairer, among many other collaborators.

We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by C*-algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc. We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.

The focus of the talk is interior-point methods for inequality-constrained quadratic programs, and particularly the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions.  We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

The Maxwell-Vlasov equations model the evolution of a plasma and can be derived from a suitably chosen Lagrangian.  That Lagrangian decomposes as the sum of terms associated with the motion of the particles, the energy stored in the electromagnetic field, and the interaction of the particles with the field.  Previous structure-preserving approaches to modeling fluid dynamics, using the group structure of the configuration space, and electromagnetic fields in vacuum, using the de Rham complex, have proved effective, raising the question whether these results could be leveraged to obtain well-behaved numerical solutions of the Maxwell-Vlasov system, thought of as a composite.  With this goal in mind, we write the Maxwell-Vlasov Euler-Poincaré equations in a weak, variational form, then use approximation spaces suggested by the referenced works to obtain a semidiscrete version of the problem.  We then present work done towards solving the fully discrete problem and indicate future directions.

The Schubert polynomials and key polynomials form two important bases for the polynomial ring. Schubert and key polynomials are the​``bottom layers” of Grothendieck and Lascoux polynomials, two inhomogeneous polynomials. In this talk, we look at their​``top layers”. We develop a diagrammatic way to compute the degrees and the leading monomials of these top layers. Finally, we describe the Hilbert series of the space spanned by these top layers, involving a classical $q$-analogue of the Bell numbers.

The aim of this talk to show that several quantities: as the spectral radius of weakly irreducible tensors, maximum of d-homogeneous polynomial with nonnegative coefficients in the unit ball of the d-H¨older norm, are polynomially computable. This computability result is proven for a larger class of minimum of certain convex functions in R n, which was considered by several authors. This is a joint work with Stephane Gaubert, INRIA and Centre de Math´ematiques Appliqu´ees (CMAP), Ecole polytechnique, IP Paris, France.

I’ll give the audience three topics that I think are interesting, and they’ll vote on which one they want to hear about. So it’s a surprise!

 A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most $R$, we obtain precise decay rates as $R$ goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.

The Whittaker Plancherel theorem appeared as Chapter 15 in my two volume book, Real Reductive Groups. It was meant to be an application of Harish-Chandra’s Plancherel Theorem.  As it turns out, there are serious gaps in the proof given in the books. At the same time as I was doing my research on the subject, Harish-Chandra was also working on it. His approach was very different from mine and appears as part of Volume 5 of his collected works; which consists of three pieces of research by Harish-Chandra that were incomplete at his death and organized and edited by Gangolli and Varadarajan. Unfortunately,  it also does not contain a proof of the theorem. There was a complication in the proof of this result that caused substantial difficulties which had to do with the image of the analog of Harish-Chandra’s method of descent. In this lecture I will explain how one can complete the proof using a recent result of Raphael Beuzzart-Plessis. I will also give an application of the result to the Fourier transforms of automorphic functions at cusps.

(This seminar will be given remotely, but there will still be a live audience in the lecture room.)

[pre-talk at 1:20PM]

The moduli spaces of one-dimensional sheaves on P^2, first studied by Simpson and Le Potier, admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one expects to obtain certain BPS invariants from the perverse filtration on cohomology induced by this morphism. This motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present a minimal set of tautological generators for the cohomology ring, and propose a “Perverse = Chern” conjecture concerning these generators, which specializes to an asymptotic product formula for refined BPS invariants of local P^2. This can be viewed as an analogue of the recently proved P=W conjecture for Hitchin systems. Based on joint work with Junliang Shen, and with Yakov Kononov and Junliang Shen.