This is an annual series of talks by a distinguished geometric analyst aimed at a general public. It is organized by Y.A. Rubinstein and S.A. Wolpert since the 2017/18 academic year.
2022-2023 Sergiu Klainerman
March 9-10, 2023
Are Black Holes Real? A Mathematical Approach to an Astrophysical Question
March 9, 2023 at 3:15pm
Sergiu Klainerman
Princeton
Abstract: The question whether black holes are real can be approached mathematically by addressing basic issues concerning their rigidity, stability and how they form in the first place. I will review these and focus on recent results concerning the nonlinear stability of slowly rotating Kerr black holes.
Presentation
Nonlinear stability of Kerr Black Holes for Small Angular Momentum
March 10, 2023 at 3:15 pm
Sergiu Klainerman
Princeton
Abstract: I will describe the main ideas behind my recent results with J. Szeftel and with E. Giorgi and J. Szeftel concerning the nonlinear stability of slowly rotating black holes.
Presentation
2018-2019 Bo Berndtsson
The classical Brunn-Minkowski theorem is an inequality for volumes of convex bodies. It can be formulated as a statement about how the volumes of vertical slices of a convex body vary when the slice varies. In these lectures we will discuss analogous results in a complex setting, where real convexity is replaced by corresponding notions in complex analysis.
Instead of slices of a convex body we then have the fibers of a holomorphic map, for instance vertical slices of a pseudoconvex domain, and instead of volumes we look at L^2-norms of holomorphic functions on the fiber. Although this picture may at first look quite different from the one in convex geometry, the Brunn-Minkowski theorem turns out to be a fruitful source of inspiration for the complex results.
October 31, 2018
Complex Brunn-Minkowski theory - Watch The Video
Wednesday, October 31 at 3:15pm
Bo Berndtsson
About the speaker
Location: Kirwan Hall 3206
Abstract: In the first lecture we will start with a gentle introduction to the Brunn-Minkowski theorem and its generalization to convex functions, Prekopa's theorem. We will then state the results in the complex setting and indicate how the real theory can be seen as a special case when we have enough symmetry. Possibly we will also give some indications of proofs and show how Hormander's L^2-estimates for the dbar-equation replaces the use of the Brascamp-Lieb inequality in the real case.
November 2, 2018
Complex Brunn-Minkowski theory - Watch The Video
Friday, November 2, at 3:15pm
Bo Berndtsson
About the speaker
Location: Kirwan Hall 3206
Abstract: In the second lecture we will turn to applications. In the first application we will give a proof of a sharp version of a famous result in complex analysis on extension with L^2-estimates of holomorphic functions defined on subvarieties of a pseudoconvex domain, the Ohsawa-Takegoshi theorem. In the proof we deform the ambient domain to a trivial case and use our theorem to show monotonicity of the constants under the deformation (joint work with L. Lempert). The next application is to the Mabuchi space of positively curved metrics on a fixed line bundle over a compact complex manifold. We will sketch a proof of a generalization of the Bando-Mabuchi uniqueness theorem for Kahler-Einstein metrics, using a complex version of Prekopa's theorem. Finally, we shall discuss applications to some positivity results from algebraic geometry, starting with a classical theorem of Griffiths.
2017-2018 Richard Schoen
March 15, 2018
Geometry and General Relativity
Thursday March 15 at 4:30pm
Richard Schoen
Stanford and UC Irvine
Abstract: This talk will be a survey of some of the geometric problems and ideas which either arose from general relativity or have direct bearing on the Einstein equations.
It is intended for a general mathematical audience with minimal physics background.
Topics will include an introduction to the Cauchy problem for the Einstein equations, problems related to gravitational mass which are closely related to the Riemannian geometry of positive scalar curvature, and trapped surfaces which are related to the mean curvature and minimal surfaces.
March 16, 2018
The Positive Mass Theorem Revisited
Friday, March 16 at 3:15pm
Richard Schoen
Stanford and UC Irvine
Abstract: We will introduce the positive mass theorem which is a problem originating in general relativity, and which turns out to be connected to important mathematical questions including the study of metrics of constant scalar curvature and the stability of minimal hypersurface singularities. We will then give a general description of our recent work with S. T. Yau on resolving the theorem on high dimensional non-spin manifolds.