In a Sea of Solutions, a Search for Uniqueness
Jiuyi Zhu, assistant professor at the LSU Department of Mathematics, recently received a $166K NSF EPSCoR Research Infrastructure Improvement grant for his work on uniqueness and quantitative uniqueness of solutions to partial differential equations. The goal of the grant is to enhance his research capacity, strengthen LSU’s research program, and benefit undergraduate and graduate education:
“My research is about uniqueness and quantitative uniqueness of solutions to partial differential equations. Math is everywhere. All phenomena in our world can be described in math terms. Partial differential equations can be used to describe and investigate numerous real-life problems, such as sound, light, heat, electrodynamics, and quantum mechanics. There can be so many solutions to these equations. Hence, it’s impossible for us to solve all of them. But if we can prove some qualitative and quantitative properties such as uniqueness and quantitative uniqueness, then we simplify the problem. If we can prove that the solution is unique, it simplifies the model significantly.
“So, what is quantitative uniqueness? It’s a way of quantifying the rate of vanishing of solutions. We want to know how a solution decays to zero. If a solution to some equation vanishes in infinite order, then the solution is trivial. But if it isn’t trivial, we want to quantify how fast the solution vanishes. Quantitative uniqueness is a developing area of research related to many problems in mathematics.”
“I do pure mathematics. There are many beautiful and important equations that uncover the mystery of nature and our world. Whereas applied research has immediate implications, what I’m doing is fundamental research. This is not like engineering or computer science where you can apply your research results to industry and get benefits directly and more or less immediately. Math is ahead of that. By working on partial differential equations, we try to understand how things work, understand nature more.
“For this project, I chose to collaborate with a world-renowned leader in the study of partial differential equations at the University of Chicago, Professor Carlos Kenig. He’s also the leading expert on uniqueness and quantitative uniqueness.
“Math is a vast subject nowadays. In our math department here at LSU, we have around 52 faculty. Each is working in his or her own area. Even mathematicians don’t necessarily understand the research other mathematicians are doing. Through the support of this grant, however, I look forward to integrating my research into my classroom teaching and increasing our research capacity overall.”
LSU Office of Research & Economic Development