题 目：Liouville Theorems and Conformally Invariant Fully Nonlinear Equations
主讲人：李岩岩 教授
单 位：美国罗格斯大学
时 间：8月1, 2日 10:00
地 点：学院南阶梯教室
摘 要：The methods of moving planes and moving spheres, together with conformal invariance, have played a significant role in the study of nonlinear elliptic equations and conformal geometry. Beginning with the classical Liouville theorem for positive harmonic functions and the celebrated classification theorem of Caffarelli, Gidas, and Spruck, the theory has evolved to encompass fully nonlinear conformally invariant equations involving the conformal Hessian (or, equivalently, the Schouten tensor). This mini-course introduces the basic structure of these equations, their conformal invariance, and their connections with the fully nonlinear Yamabe problem. We will discuss recent progress on Liouville-type theorems for general conformally invariant fully nonlinear equations, including sharp necessary and sufficient conditions for their validity. Particular emphasis will be placed on isolated singularities and the optimal geometric conditions on admissible cones. Time permitting, we will also discuss applications to local gradient estimates and problems in conformal geometry. The course is intended for graduate students and researchers with a background in elliptic partial differential equations and differential geometry.
简 介： Yanyan Li is a Distinguished Professor at Rutgers University and the Director of the Center for Nonlinear Analysis. He received his B.S. degree from the University of Science and Technology of China in 1982, his M.S. from the Institute of Systems Science, Academia Sinica in 1983, and his Ph.D. from the Courant Institute of Mathematical Sciences at New York University in 1988. His research interests lie in nonlinear partial differential equations and their applications. He was an invited speaker at the International Congress of Mathematicians in 2002 and has been a member of the inaugural class of Fellows of the American Mathematical Society since 2012. He was awarded an Alfred P. Sloan Research Fellowship (1993–1995), a Simons Fellowship in Mathematics and Theoretical Physics in 2020, and the Rutgers Board of Trustees Award for Excellence in Research in 2008.
