Scientific

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.

In longitudinal studies, we measure the same variables at multiple time-points to track their change over time. The exact data collection schedules (i.e., time of participants' visits) are often pre-determined to accommodate the ease of project management and compliance. Therefore, it is common to schedule those visits at equally spaced time intervals. However, recent publications based on simulated experiments indicate that the power of studies and the precision of model parameter estimators is related to the participants' visiting scheme. So, in this work, we investigate how to schedule participants' visits to better study the accelerated cognitive decline of senior adults, where a broken-stick model is often applied. We formulate this optimal design problem on scheduling participants' visiting into a high- dimensional optimization problem and derive its approximate solution by adding reasonable constraints. Based on this approximation, we propose a novel design of the visiting scheme that aims to maximize the power (i.e. reduce the variance of estimators) in identifying the onset of accelerated decline. Using both simulation studies and evidence from real data, we demonstrate that our design outperforms the standard equally-spaced one when we have strong prior knowledge on the change-points. This novel design helps researchers plan their longitudinal studies with improved power in detecting pattern change without collecting extra data. Also, this individual-level scheduling system helps monitor seniors' cognitive function and, therefore, benefits the development of personal level treatment for cognitive decline, which agrees with the trend of the health care system.

Synthetic biology offers bottom-up engineering strategies that intends to understand complex systems via design-build-test cycles. In development, gene regulatory networks emerge into collective cellular behaviors with multicellular forms and functions. Here, I will introduce a synthetic developmental biology approach for tissue engineering. It involves building developmental trajectories in stem cells via programmed gene circuits and network analysis. The outcome of our approach is decoding our own development and to create programmable organoids with both natural or artificial designs and augmented functions.