Groupoids give a very large class of examples of C*-algebras. For example, it is known that every classifiable C*-algebra arises as the reduced C*-algebra of some twisted groupoid.

In joint work with Matthew Kennedy, Se-Jin Kim, Xin Li, and Sven Raum, we fully characterize when the essential C*-algebra of an étale groupoid $\mathcal{G}$ with locally compact unit space has the ideal intersection property. This is done in terms of the dynamics of $\mathcal{G}$ on the space of subgroups of the isotropy groups of $\mathcal{G}$. The essential and reduced C*-algebras coincide in the case of Hausdorff groupoids, and the ideal intersection property is the same as simplicity in the case of minimal groupoids. This generalizes the case of the reduced crossed product $C(X) \rtimes_r G$ done by Kawabe, which in turn generalizes the case of the reduced C*-algebra $C^*_r(G)$ of a discrete group done by Breuillard, Kalantar, Kennedy, and Ozawa.

No prior knowledge of groupoids will be required for this talk.

I will communicate an amusing observation about the K theory of non-noetherian schemes in characteristic zero. The lowest K group in this setting can sometimes identify with the top cohomology of the structure sheaf. No knowledge of negative K theory (or even K theory!) will be assumed, with the hope that both topologists and algebraic geometers can learn something.

Stochastic iterative methods lie at the core of large-scale optimization and its modern applications to data science. Though such algorithms are routinely and successfully used in practice on highly irregular problems (e.g. deep neural networks), few performance guarantees are available outside of smooth or convex settings. In this talk, I will describe a framework for designing and analyzing stochastic gradient-type methods on a large class of nonsmooth and nonconvex problems. The problem class subsumes such important tasks as matrix completion, robust PCA, and minimization of risk measures, while the methods include stochastic subgradient, Gauss-Newton, and proximal point iterations. I will describe a number of results, including finite-time efficiency estimates, avoidance of extraneous saddle points, and asymptotic normality of averaged iterates.

Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, "Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?"

We'll discuss recent work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We'll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities.

In joint work with Lucas Mann, we establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we prove a duality theorem, establish bounds on Gelfand-Kirillov dimension, prove some non-vanishing results, and show a kind of partial Künneth formula. The key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by Mann.

The aim of this talk is to introduce the audience to the Halpern-Lauchli theorem, which is a Ramsey-theoretic statement about products of trees. We will discuss several applications of the theorem and outline a number of different proofs. While the original proof was combinatorial in nature, there are now a number of proofs that interact with ideas from set-theoretic forcing. One of these proofs is new, and is joint work with Chris Lambie-Hanson.

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.