We show that, for many choices of finite tuples of generators $\mathbf{X}=(x_1,\dots,x_d)$ of a tracial von Neumann algebra $(M,\tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $\Gamma$), one can find a diffuse, hyperfinite subalgebra in $W^*(\mathbf{X})^\omega$ (often in $W^*(\mathbf{X})$ itself), such that $W^*(N,\mathbf{X}+\sqrt{t}\mathbf{S})=W^*(N,\mathbf{X},\mathbf{S})$ for all $t>0$. (Here $\mathbf{S}$ is a free semicircular family, free from $\{\mathbf{X}\cup N\}$). This gives a short 'non-microstates' proof of strong 1-boundedness for such algebras.

This is joint work with Dimitri Shlyakhtenko.

In this talk, I will discuss a new physical-space approach to establishing the time decay and global asymptotics of solutions to variable-coefficient Schrödinger equation in (3+1)-dimensions. The result is applicable to possibly large, time-dependent, complex-valued coefficients under a general set of hypotheses. As an application, we are able to handle certain quasilinear cubic and Hartree-type nonlinearities, proving global existence together with global asymptotics. I will begin with a model problem and describe the construction of a good commutator. Time permitting, I will explain how to incorporate the good commutator with Ifrim--Tataru the method of testing by wave packets to obtain global asymptotics. This talk is based on upcoming work with Sung-Jin Oh and Federico Pasqualotto.

That is the question. In this talk, I will describe several problems of varying degrees of “tubiness” (the amenability of the problem to tube technology).

The number of regions of a hyperplane arrangement is a well-understood invariant, which we can complicate by counting regions of a given \emph{level}, a statistic quantifying a region's boundedness. Rediscovering a formula of Zaslavsky, we show that the level distribution is a \emph{combinatorial invariant}, and in the process define it for all semimatroids. The formula also allows us to reprove and generalize many known results on deformations of the braid arrangement.

Gaussian multiplicative chaos (GMC) is a well-studied random measure appearing as a universal object in the study of Gaussian or approximately Gaussian log-correlated fields. On the other hand, no general framework exists for the study of multiplicative chaos associated to non-Gaussian log-correlated fields. In this talk, we examine a canonical model: the log-correlated random Fourier series, or random wave model, with i.i.d. random coefficients taken from a general class of distributions. The associated multiplicative chaos measure was shown to be non-degenerate when the inverse temperature is subcritical ($\gamma < \sqrt{2d}$) by Junnila. The resulting chaos is easily seen to not be a GMC in general, leaving open the question of what properties are shared between this non-Gaussian chaos and GMC. We answer this question through the lens of absolute continuity, showing that there exists a coupling between this chaos and a GMC such that the two are almost surely mutually absolutely continuous.

This talk will present new results on the optimal control of self-organization, motivated by a growing body of empirical work on biological pattern formation in dynamic environments. We pose a boundary control problem for the classic supercritical Turing pattern, asking the best way to reach a non-trivial steady state by controlling the boundary flux of a reactant species. Via the Pontryagin approach, first-order optimality conditions for a generic reaction-diffusion system with a suitable bifurcation structure are derived. Using formal asymptotics, we construct approximate closed-form optimal solutions in feedback law form that are valid for any Turing-unstable system near criticality, which are verified against numerical solutions for a representative reaction-diffusion model.

Algebraic curves and abelian varieties play a central role in modern algebraic geometry, with links to complex analysis, number theory, topology and others. Curves and abelian varieties are closely related: a fundamental example of an abelian variety is the Jacobian of an algebraic curve. In this talk, I will give a discussion of curves, abelian varieties and their moduli spaces. Time permitting, I will present some new tools aimed at studying geometric classes on the moduli space of abelian varieties, and conclude with a discussion of several open questions in this area.