Title: Nonlinear Elliptic Partial Differential Equations in Geometry
Speaker: Bo Guan
Abstract: Partial differential equations play very important roles in geometry as well as other fields of science and technology. In fact, using PDE methods to solve difficult problems in geometry has been a great achievement in mathematics since the mid-twentieth century, marked by the proofs of Bernstein Theorem, Calabi conjecture, Positive Mass Theorem in general relativity, Yamabe problem, and Poincare conjecture, to name a few.
In our talk we will start with some simple applications of PDE methods to geometric problem, including a proof of the isoperimetric inequality, Alexanderov's theorem, and if time permits a proof of Sobolev inequality using Monge-Ampere equation. We then present some classical problems in geometry and related fully nonlinear PDEs, such as Weyl problem of isometric embeddings, Minkowski problem and extensions by Alexanderov and Chern, Calabi conjecture, and Donaldson conjecture, etc. Hopefully we can also describe the famous Plateau problem which was first raised by Lagrange in 1760, and solved in 1930's independently by Jesse Douglas who received the first Fields medal in 1936 because of this work, and the late OSU professor Tibre Rado. Along the line we hope to be able to briefly mention some of our own related work.
At the last part of our talk we will try to give some idea about our more recent work on solving fully nonlinear elliptic and parabolic equations on general Riemannian and complex manifolds.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy PhD students can sign up for this lecture series by registering for two credit hours of Math 6193 with Professor Nimish Shah.
