Prob/Stat Seminar

The coordinate functions on a free boundary minimal surface (FBMS) in the unit ball B^n are Steklov eigenfunctions with eigenvalue 1. For many embedded FBMS in B^3 we show its first Steklov eigenspace coincides with the span of its coordinate functions, affirming a conjecture of Fraser & Li in an even stronger form.  One corollary is a partial resolution of the Fraser-Schoen conjecture: the critical catenoid is the unique embedded FBM annulus in B^3 with antipodal symmetry. [This is joint work with Peter McGrath.]

Percolation clusters induce an intrinsic graph distance called the chemical distance. Besides its mathematical appeal, the chemical distance is connected to the study of random walks on critical percolation clusters. In this talk, I will begin with a brief survey on the chemical distance. Then, I will zoom in to the progress and challenges in the 2d critical regime. A portion of this talk is based on joint work with Philippe Sosoe.

Branching Brownian motion is a random particle system that incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom type results. Specifically, we will discuss the construction of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. Based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg.

In this talk we will consider small deviations (SD) and their connection to spectral gaps and Chung’s laws of the iterated logarithm (LIL). The main focus is on hypoelliptic diffusions such as the Kolmogorov diffusion and horizontal Brownian motions on Carnot groups. We will also discuss spectral properties and existence of spectral gaps for hypoelliptic operators. This talk is partially based on some works with Maria Gordina.

In this talk I will discuss some large deviation results coming from super-critical Branching Diffusion Processes in d-dimensional space. These results give precise asymototics for the density of particles in the domain large deviations thus enabling us to obtain regions in space-time where intermittency for k-th moments occurs. I will also talk about higher order asymptotics for large deviation probabilities of general stochastic processes which have weakly dependent increments.

Abelian networks a class of models from statistical physics introduced as models to help understand the complex behavior exhibited by forest fires and avalanches. In this talk we will introduce a few different Abelian networks including activated random walk and the stochastic sandpile model. We will discuss the techniques used to study them and discuss some recent results.

Investigating the differential effect of treatments in groups defined by patient characteristics is of paramount importance in personalized medicine research. In some studies, participants are first classified as having or not of the characteristic of interest by diagnostic tools, but such classifiers may not be perfectly accurate. The impact of diagnostic misclassification in statistical inference has been recently investigated in parametric model contexts and shown to introduce severe bias in estimating treatment effects and give grossly inaccurate inferences. In this talk, we address these problems in a fully nonparametric setting. Methods for consistently estimating and testing meaningful yet nonparametric treatment effects are developed. The applications of the proposed methods are illustrated with gene expression profiling of bronchial airway brushing in asthmatic and healthy control subjects.