Reaction-diffusion equations with nonlocal constraints naturally arise as limiting cases of mathematical models of intracellular signaling. Among the interesting behaviors of these models, much has been made of their 'geometry-sensing' properties: the strong sensitivity of steady-state solutions to domain geometry is widely seen as illustrative of how a cell establishes an internal coordinate axis. In this talk, I describe recent efforts to formally clarify this geometry dependence through careful study of the long-time behavior of a popular model of biochemical symmetry breaking.  Using the tools of formal asymptotics, calculus of variations, and a new fast solver for surface-bound PDEs, we study the formation and motion of interfaces on a curved domain across three dynamical timescales. Our results allow us to construct several analytical steady-state solutions that serve as counter-examples to received wisdom regarding the geometry-dependence of this class of model. 

From social media trends to family dynamics, social interactions shape our daily lives. In this talk, I will present tools I have developed for statistical inference and decision-making in light of these social interactions.

(1) Inference: I will talk about estimation of causal effects in the presence of interference. In causal inference, the term “interference” refers to a situation where, due to interactions between units, the treatment assigned to one unit affects the observed outcomes of others. I will discuss large-sample asymptotics for treatment effect estimation under network interference where the interference graph is a random draw from a graphon. When targeting the direct effect, we show that popular estimators in our setting are considerably more accurate than existing results suggest. Meanwhile, when targeting the indirect effect, we propose a consistent estimator in a setting where no other consistent estimators are currently available.

(2) Decision-Making: Turning to reinforcement learning amid social interactions, I will focus on a problem inspired by a specific class of mobile health trials involving both target individuals and their care partners. These trials feature two types of interventions: those targeting individuals directly and those aimed at improving the relationship between the individual and their care partner. I will present an online reinforcement learning algorithm designed to personalize the delivery of these interventions. The algorithm's effectiveness is demonstrated through simulation studies conducted on a realistic test bed, which was constructed using data from a prior mobile health study. The proposed algorithm will be implemented in the ADAPTS HCT clinical trial, which seeks to improve medication adherence among adolescents undergoing allogeneic hematopoietic stem cell transplantation.

Many statistical learning problems, where one aims to recover an underlying low-dimensional signal, are based on optimization, e.g., the linear programming approach for recovering a sparse vector. Existing work often either overlooked the high computational cost in solving the optimization problem, or required case-specific algorithm and analysis -- especially for nonconvex problems. This talk addresses the above two issues from a unified perspective of conditioning. In particular, we show that once the sample size exceeds the intrinsic dimension of the signal, (1) a broad range of convex problems and a set of key nonsmooth nonconvex problems are well-conditioned, (2) well-conditioning, in turn, inspires new algorithm designs and ensures the efficiency of many off-the-shelf optimization methods.