Summer School in Geometric Analysis

Northwestern University
July 6-12, 2015


All lectures to be held in Lunt Hall 105. Click here for a campus map.

Registration and coffee starts at 8.30am on Monday, July 6, in Lunt Hall 105

Monday, July 6 Tuesday, July 7 Wednesday, July 8 Thursday, July 9 Friday, July 10 Saturday, July 11 Sunday, July 12
9.00am - 10.30 am Weinkove Weinkove Kotschwar Weinkove Naber Cabezas-Rivas Tosatti
11.00am - 12.30 pm Naber Cabezas-Rivas Cabezas-Rivas Zelditch Zelditch Tosatti Zelditch
2.00pm - 3.30 pm Kotschwar Naber Free afternoon Naber Kotschwar Zelditch
4.00pm - 5.30 pm Weinkove Kotschwar Free afternoon Cabezas-Rivas Tosatti Tosatti


Monday July 6, 5.30pm there will be Pizza and Beer in the Mathematics Department Common Room, 2nd floor Lunt Hall

Saturday July 11, 5.30pm there will be a Wine and Cheese reception in the Mathematics Department Common Room, 2nd floor Lunt Hall



Esther Cabezas-Rivas   (Frankfurt). Title: Mean Curvature Flow

Abstract: Non-linear heat equations have played an important role in differential geometry and topology over the last decades. Broadly speaking, a geometric quantity or structure on a manifold is evolved in a canonical way towards an optimal one, that is, we deform a manifold into another one with nicer properties. During the lectures we will focus on the curve-shortening flow and its higher dimensional analogue, the mean curvature flow, which deform a curve (hypersurface) in its normal direction with speed equal to the curvature (mean curvature) at each point. Analytically, this process is described by a weakly parabolic system of partial differential equations for the local embedding map of the evolving hypersurfaces. At the curvature level it looks like a reaction-diffusion system. The reaction part, which is cubic in the curvatures, generally forces the formation of singularities (points near which the curvature blows up) in finite time. The diffusion part, involving the Laplace-Beltrami operator of the moving hypersurface, shares many properties with the heat equation; in particular, it tends to balance differences e.g. of the curvature on the manifold (so only with the diffusion effect the curvature will eventually tend to a constant). As these two effects are competing, we need a combination of techniques of analysis and geometry to control the behaviour of the flow. During the lectures we will introduce the basic techniques and provide proofs of some of the classical results in the field.

Brett Kotschwar   (Arizona State University). Title: Introduction to the Ricci flow

Abstract: These lectures are meant to provide a first look at the Ricci flow for students with some prior coursework in differential geometry. After considering some special classes of solutions, we will discuss the existence and uniqueness of solutions and other fundamental analytic properties of the equation. We will then consider the application of various versions of the maximum principle to the evolution of curvature quantities under the flow and explore a few consequences for the long-time behavior of solutions. We will also discuss some basic techniques in singularity analysis and combine them with a curvature pinching estimate to give a synthetic proof of Hamilton's 1982 theorem on closed three-manifolds of positive Ricci curvature.

Aaron Naber   (Northwestern). Title: Ricci curvature

Abstract: This course will discuss various aspects of analysis on Riemannian manifolds, focusing in particular on the role of Ricci curvature. The first part of the course will discuss the heat flow on Riemannian manifolds with lower Ricci curvature bounds, and in particular the Bakry-Emery-Ledoux estimates, the Li-Yau Harnack inequalities, with applications to the heat kernel. The last couple days of the course will discuss more recent structural developments in the field, including an overview of the regularity theory of Einstein manifolds.

Valentino Tosatti   (Northwestern). Title: The Kähler-Ricci flow

Lecture notes

Abstract: The Ricci flow is an evolution equation which deforms a Riemannian metric in the direction of its Ricci tensor. If the underlying manifold is complex and the initial metric is Kähler then so are the evolved metrics, and the flow is called the Kähler-Ricci flow. When the manifold is also compact, the flow becomes intimately related to the complex structure of the manifold, and if the manifold is algebraic the convergence properties of the flow are directly related to the minimal model program in birational geometry. In these lectures I will give an introduction to the Kähler-Ricci flow, and present some results which fit in this framework. The topics to be covered are: the characterization of the maximal existence time of the flow, the formation of singularities in finite time, the long time behavior on minimal Kähler manifolds, and (time permitting) the case of Kähler surfaces where the picture is essentially complete.

Ben Weinkove   (Northwestern). Title: Introduction to complex geometry and the complex Monge-Ampère equation

Abstract: I will begin with a brief introduction to complex and Kähler geometry, assuming only a basic knowledge of differential geometry and complex analysis. There will be an emphasis on calculations and simple examples. I will then introduce the complex Monge-Ampère equation on compact Kähler manifolds, and describe Yau's proof of existence of solutions. I will explain how this implies the existence of Calabi-Yau metrics on certain manifolds.

Steve Zelditch   (Northwestern). Title: Geodesics in the space of Kähler metrics

Supplementary material

Abstract: Hele-Shaw flow is a geometric flow on domains in which the normal velocity of the boundary at a point equals the harmonic measure at that point. As the domain evolves its area grows linearly in time but all other harmonic moments are constant. Hele-Shaw flow is closely related to the motion of equlibrium measures as the potential is increased. In work of Wiegmann-Zabrodin, it is related to random matrices and the quantum Hall effect. In other recent work of Ross and Witt-Nystrom it has also been related to geodesics in the space of Kaehler metrics and to holomorphic discs. Despite these unexpected and far-ranging connections, the Hele-Shaw flow is quite elementary and can be understood with only a first course in complex analysis. The only prerequisite is the Riemann mapping theorem and the use of the Perron envelope method for solving the Dirichlet problem and constructing Green's functions (we will review the latter).