CAREER: Fluctuation estimate, Selection principle and Transition paths for multiscale interacting dynamics on complex structure

Interactive dynamics, where individuals or particles influence each other and the overall system, are important in fields like biology, physics, materials science, and social sciences. These dynamics often involve rare events that can have large impacts, such as changes in protein structures or genetic evolution. Inaccurate predictions of these events can hinder our understanding of biological processes, drug design, and our ability to prepare for extreme events.

This project seeks to improve our ability to predict and control these rare but critical events by identifying key patterns in energy landscapes and developing new methods for estimating fluctuations and controlling extreme events in complex spaces. The insights gained will not only help in predicting extreme events but also improve our understanding of various interactive behaviors, such as material design and social opinions.

Educational initiatives will promote interdisciplinary learning and advance the growth of applied mathematics and related fields.

This project aims to predict and control rare, significant events in interactive dynamics on complex configuration spaces. The goal is to advance the theoretical understanding of Hamilton-Jacobi equations (HJEs) and control theory for multiscale interactions by addressing key challenges, including non-uniqueness of stationary solutions, fluctuation estimates in multiscale interactions, and singular optimal control in probability spaces.

Theoretical developments will be applied to practical problems, such as rare event simulations, to enable more predictable dynamics in complex systems. First, the investigator will develop a systematic method to select stationary solutions to HJEs, providing a uniformly converging vanishing viscosity approximation and a global energy landscape. Second, a novel decomposition for species concentration and reaction fluxes in non-equilibrium multiscale reactions will be introduced, along with a singular limit framework for fluctuation estimates.

Third, rigorous justifications for the singular limit of variational solutions for HJEs with state constraints will be achieved via the equivalence between optimal control and optimal path measures for transition problems in infinite dimensional spaces, particularly in the presence of boundary singularities. The research outcomes will be disseminated through conferences, publications, and online platforms.

The approaches developed will also be integrated into undergraduate and graduate teaching, offering research opportunities for graduate students.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.