Dr. Fu’s research interest is spectral analysis of partial differential operators and geometric analysis in several complex variables. The basic theme is to study the interplay among geometric, analytic, and algebraic structures of a complex manifold. His research interest also includes invariant metrics and automorphism groups on bounded domains in several complex variables. More recently, Dr. Fu has been working on spectral theory of complex Laplacians and its application to problems in complex and algebraic geometries. He is also interested in the connection between mathematics and music.
Complex analysis of several variables is a branch of mathematics where analysis, algebra, and geometry intertwine. Complex analysis is an essential tool in physics and engineering. It have been applied to study heat diffusion and fluid dynamics by physicists and engineers for more than two centuries. Spectral theory of differential operators plays an important role in many areas of biological, medical, and physical sciences. For example, the inverse problem–the problem of determining the shape from spectral properties–has applications in fields such as medical imaging. Pure discreteness of the spectra of the complex Laplacians is intimately related to diamagnetism and paramagnetism, topics widely studied in quantum physics and chemistry.
Dr. Fu has received research grants from the National Science Foundation with funding supports for undergraduate and graduate students. He has supervised and mentored a number students, including a high school student. Some of these students went on to pursue Ph.D. degrees in related fields while others found employments in education, finance, industry, and government agencies.
Professor Fu received his Ph.D. degree in Mathematics from Washington University in St.Louis. He was a recipient of an American Mathematical Society Centennial Research Fellowship in 2000.
Selected Publications:
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Compactness of the
-Neumann problem on convex domains (with E. Straube), Journal of Functional Analysis 159 (1998), 629-641. -
Compactness in the
-Neumann problem, magnetic Schrödinger operators, and the Aharonov-Bohm effect (with Michael Christ), Advances in Mathematics 197 (2005), 1-40. -
Hearing pseudoconvexity with the Kohn Laplacian, Mathematische Annalen 331 (2005), 475-485.
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Hearing the type of a domain in C2 with the
-Neumann Laplacian, Advances in Mathematics 219 (2008), 568-603. -
Comparison of the Bergman and Szegö kernels (with Bo-Yong Chen), Advances in Mathematics 228 (2011), 2366-2384.
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Stability of the Bergman kernel on a tower of coverings (with Bo-Yong Chen), Journal of Differential Geometry 104 (2016), 371-398.
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Hearing pseudoconvexity in Lipschitz domains with holes via
(with Christine Laurent-Thiébaut and Mei-Chi Shaw), Mathematische Zeitschrift 287(2017), 1157-1181. -
Spectral stability of the
-Neumann Laplacian: Domain perturbations (with Weixia Zhu), Journal of Geometric Analysis 32 (2022), 1-34.
Teaching:
Probability and Stochastic Processes (56:645:558/56:219:512)
Applied Partial Differential Equations/Method of Applied Mathematics (56:640:463/56:645:527)