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.

### 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.

### 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.