Scientific

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable convexity/dissipativity assumptions, we show the exponential ergodicity for both systems, establish a uniform-in-time law of large numbers for the empirical measure of particle states, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided. We also provide uniform-in-time exponential concentration bounds for the rate of the LLN convergence under additional integrability conditions. Based on joint works with Erhan Bayraktar and Suman Chakraborty.

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable additional assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

Based on joint works with Suman Chakraborty and Ruoyu Wu.

We coexist with a vast number of microbes—our microbiota—that live in and on our bodies, and play an important role in human physiology and diseases. Many scientific advances have been made through the work of large-scale, consortium-driven metagenomic projects. Despite these advances, there are still many fundamental questions regarding the dynamics and control of microbiota to be addressed. Indeed, it is well established that human-associated microbes form a very complex and dynamic ecosystem, which can be altered by drastic diet change, medical interventions, and many other factors. The alterability of our microbiome offers opportunities for practical microbiome-based therapies, e.g., fecal microbiota transplantation and probiotic administration, to restore or maintain our healthy microbiota. Yet, the complex structure and dynamics of the underlying ecosystem render the quantitative study of microbiome-based therapies extremely difficult. In this talk, I will discuss our recent theoretical progress on controlling human microbiota from network science, dynamical systems, and control theory perspectives.

Protocols and devices that exploit quantum mechanical effects can outperform their classical counterparts in certain tasks ranging from communication and computation to sensing. Intuitively speaking, the reason for this is that different physical laws allow for different technological applications. Therefore, the question where quantum mechanics differs from classical physics is not only of foundational or philosophical interest but might have technological implications too. To address it in a systematic manner, so-called quantum resource theories were developed. These are mathematical frameworks that emerge from (physically motivated) restrictions that are put on top of the laws of quantum mechanics and single out specific aspects of quantum theory as resources. A widely studied example would be the restriction to local operations and classical communication, which leads to the resource theory of entanglement. It is then investigated how these restrictions influence our abilities to do certain tasks (e.g., communicate securely), how these restrictions can be overcome, and how the resulting resources can be quantified. Historically, resource theories were mainly focused on the resources present in quantum states. After an introduction to the general topic, I will speak about my recent research on how these concepts can be extended to quantum operations and why this is of interest.

Recursive distributional equations (RDEs) are ubiquitous in probability. For example, the standard Gaussian distribution can be characterized as the unique fixed point of the following RDE

$$
X = (X_1 + X_2) / \sqrt{2}
$$

among the class of centered random variables with standard deviation of 1. (The equality in the equation is in distribution; the random variables and must all be identically distributed; and and must be independent.)

Recently, it has been discovered that the dynamics of certain recursive distributional equations can be solved using by using tools from numerical analysis, on the convergence of approximation schemes for PDEs. In particular, the framework for studying stability and convergence for viscosity solutions of nonlinear second order equations, due to Crandall-Lions, Barles-Souganidis, and others, can be used to prove distributional convergence for certain families of RDEs, which can be interpreted as tree- valued stochastic processes. I will survey some of these results, as well as the (current) limitations of the method, and our hope for further interplay between these two research areas.