Speaker：Prof. Lei Ni（倪磊）（University of California, San Diego）
Time：20, December, 14:30--15:30
Geometry is an old subject whose systematical study was started by ancient Greeks. The geometry that students learn today in the middle school are mostly contained in Euclid’s Elements. Modern geometry studies the geometric, topological properties of manifolds. A very important concept is the curvature of a Riemannian manifold. Even at the time of Gauss, the global metric and topological implication of the curvature was investigated. It later played a fundamental role in Einstein’s theory of the general relativity (see for example, Gravity by Misner, Thorne and Wheeler). I pick two important theorems from geometry. One is the Gauss-Bonnet theorem, which is covered in somewhat standard undergraduate geometry texts (such as Do Carmo’s Differential geometry of curves and surfaces). The other is the Toponogov’s comparison theorem, an important tool responsible for the development of the Riemannian geometry for more than twenties years since the middle 1960s (see for example, the textbook Comparison theorems in Riemannian geometry by Cheeger and Ebin), which is often taught in a graduate course on Riemannian geometry. In this talk I shall illustrate the connections between these two results and two elementary results from the middle school geometry.
Differential analysis has its important role since the beginning of the Riemannian geometry. For example it allows one to formulate general results such as the Gauss-Bonnet theorem. Since the early part of the last century the PDE method slowly becomes a more dominating tool in the study of the differential geometry. Geometric analysis is a subject which utilizes the analytic tools, mainly the theory of the partial differential equations, to study the geometric, topological and analytic properties of manifolds. One of the most effective tools of the PDE theory is the maximum principle. In this talk I shall show how one can derive the Toponogov’s theorem out of a maximum principle. The proof employed is an improvement of the approach discovered by Karcher. This drastically simplifies the standard proof, for example the one in the previous mentioned textbook of Cheeger and Ebin. Spectrum properties of partial differential operators contain much geometric information of the underlying Riemannian manifolds. As another application of the maximum principle we illustrate a method, which is motivated by corresponding results from the theory of matrices (mostly taught in a college algebra course), of proving a fundamental result concerning the principle eigenvalue of a nonsymmetric second order elliptic partial differential operator.