In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimeswith odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-timetailson stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-timetailsare in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of the problem. This is joint work with Jonathan Luk(Stanford).

In this talk, I will discuss some recent developments on the long-term dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariantsymmetry. CSS is a gauge-covariant 2D cubic nonlinear Schrödingerequation, which admits the L2-scaling/pseudoconformalinvariance and soliton solutions.I will first discuss soliton resolution for this model, which is a remarkable consequence of the self-duality and non-local nonlinearity that are distinguished features of CSS. Next, I will discuss the blow-up dynamics (singularity formation) for CSS and introduce an interesting instability mechanism (rotational instability) of finite-time blow-up solutions. This talk is based on joint works with SoonsikKwon and Sung-JinOh.

In this talk, we discuss the fluctuation of f(X) as a matrix, where  X is a large square random matrix with  centered, independent, identically distributed entries and f is an analytic function. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two independent modes corresponding to tracial and traceless parts of A. We also briefly discuss the proof that mainly relies on Hermitization of X and its resolvents. 

We consider an optimal transport problem where the cost depends on the stopping time of Brownian motion from a given distribution to another. When the target measure is fixed, it is often called the optimal Skorokhod embedding problem in the literature, a popular topic in math finance. Under a monotonicity assumption on the cost, the optimal stopping time is given by the hitting time to a space-time barrier set. When the target measure is optimized under an upper bound constraint, we will show that the optimal barrier set leads us to the Stefan problem, a free boundary problem for the heat equation describing phase transition between water and ice. This is joint work with Young-Heon Kim at UBC.

In this talk, we consider the Ising and Potts model defined on large lattices of dimension two or three at very low temperature regime. Under this regime, each monochromatic spin configuration is metastable in that exit from the energetic valley around that configuration is exponentially difficult. It is well-known that, under the presence of external magnetic fields, the metastable transition from a monochromatic configuration to another one is characterized solely by the appearance of a critical droplet. On the other hand, for the model without external field, the saddle structure is no longer characterized by a sharp droplet but has a huge and complex plateau structure. In this talk, we explain our recent research on the analysis of this energy landscape and its application to the demonstration of Eyring-Kramers formula for models on fixed two or three dimensional lattice (cf. https://arxiv.org/abs/2102.05565) or models on growing two-dimensional lattice (cf. https://arxiv.org/abs/2109.13583).

In this talk, I will describe the large deviation asymptotic of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in the Poisson $k$-nearest neighbor graph in $\mathbb R^d$. While the case $\alpha < d$ can be treated through classical methods from large deviations theory, an interesting dichotomy occurs if $\alpha > d$. Rare events in the lower tail can still be explained by subtle changes in the Poisson process throughout the sampling window. However, the most likely cause for rare events in the upper tail is a condensation phenomenon: the excess edge weight is caused by a negligible portion of Poisson points whose configuration can be described through a concrete geometric optimization problem. After presenting the general proof strategy, I will also elucidate on the prospects and limits of generalizing our approach to other spatial networks.