Topics in Wave Propagation and Quantum Systems

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Wave phenomena are ubiquitous in nature, and are virtually in every aspect of every day life. They are responsible for the sound we hear, the light we see, and the data we receive on our personal devices. Understanding how to mathematically model these waves, how to control them, and how to extract valuable information from them is therefore an important question at the intersection of various fields. The underlying physics principles behind wave propagation are extremely complex, and mathematics has proven to be a crucial tool in their investigation. At the core of this project are the theoretical analysis and the simulation of wave phenomena, with an emphasis on three main problems: how to focus energy with a minimal experimental apparatus; how to infer physical properties of a complex medium using wave propagation; and how to model waves in novel quantum devices. This project integrates research with the training of the next generation of scientists in the field of applied mathematics.The main theme of research of the project is wave phenomena, including both classical and quantum waves. There are three main components to the proposed research. The first one originates from the recent groundbreaking experiments on instantaneous time mirrors. The latter form a novel avenue for time reversal and the control of waves, and is an exciting new topic for mathematicians to explore. The second component concerns inverse problems based on stochastic correctors, which are considered physical systems where forward data are solutions to PDEs with small parameters and random coefficients. In these systems, the data are asymptotically modeled by random correctors to a leading term that is not accessible to measurements. The objectives are first to characterize these stochastic correctors, and second to develop inversion strategies based on these corrections to extract information on the system. A typical example is the “sea ice problem”, which involves random fluctuations around a homogenized solution. The last component of this project concerns quantum systems. The first problem is to develop the mathematical foundations of quantum hydrodynamical models based on the entropy principle, and the second the derivation of efficient numerical schemes for periodic Dirac operators, with applications to the study of exotic phases of matter and the design of novel quantum devices.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.