Geometric Analysis

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

The 23nd Annual PCMI Summer Session will be held June 30 – July 20, 2013.

Michael Eichmair: On the isoperimetric structure of asymptotically flat manifolds

In this series of lectures we discuss some fascinating connections between the isoperimetric structure of an asymptotically flat three-manifold, its scalar curvature, and the physical properties of the space-time evolving from such a geometry when viewed as initial data for the Einstein equations. We will begin with a survey of the minimal hypersurface proof of the positive mass theorem (due to Schoen and Yau) and results on the existence of canonical foliations by stable constant mean curvature surfaces (due to Huisken and Yau). We will then connect with recent work that re-interprets these results in terms of the solution to the isoperimetric problem for large volumes.

Fernando Marques: "Min-max theory and the Willmore conjecture"

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of any torus immersed in Euclidean three-space is at least 2\pi^2. In this minicourse, I intend to discuss recent joint work with Andre Neves in which we prove this conjecture using the min-max theory of minimal surfaces.

Tristian Riviere: Weak immersions of surfaces with finite total curvature''

''A weak immersion  with finite total curvature of a surface is a Lipschitz immersion whose Gauss Map has a finite Dirichlet Energy.  The formalism of rectifiable cycles has been originally introduced to study the variations of the area functional.  It is a notion which has been shaped around the finite area constraint and for which a closedness theorem is available...etc. In a similar vein, the notion of weak immersion of surfaces with finite total curvature is adapted to the study of the higher order Lagrangian given by the $L^2$ norm of the second fundamental form. We shall give an almost closedness theorem for the space of weak immersions of finite total curvature and explain why this notion is the weakest framework which provides well defined smooth underlying conformal structure for immersion of surfaces. The variations of the $L^2-$norm of the second fundamental form, or the Willmore Lagrangian for closed 2-manifold, under conformal class constraints, is generating families of surfaces with rich analysis and geometric features that we shall present''

Igor Rodnianski: "Evolution problem in General Relativity"

The lectures will review some of the classical topics in the subject of the Cauchy problem for the Einstein equations, including its formulation, geometric description of solutions and focusing phenomena encapsulated in the Penrose incompleteness theorem. They will also address more recent developments on the well-posedness and stability problems, including L^2 curvature conjecture, impulsive gravitational waves and trapped surface formation.

Peter Topping: "Applications of Hamilton's Compactness Theorem for Ricci flow"

In these lectures, I will try to give an introduction to two separate aspects of Ricci flow, namely Hamilton’s Compactness Theorem and the very neat theory of Ricci flow in 2D. The target audience consists of graduate students with some background in differential geometry and PDE theory.

Hamilton’s Compactness Theorem is an absolutely fundamental tool in the modern theory of Ricci flow. I will spend the early part of the course explaining what this result says – roughly that given an appropriate sequence of Ricci flows, one can pass to a subsequence and get smooth convergence to a limit Ricci flow in the `Cheeger-Gromov' sense. We will not assume any prior knowledge of this notion of convergence, although it will be almost essential to have some basic prior knowledge of Riemannian geometry, including the basic idea of the Riemannian curvature tensor.

I will then go on to illustrate the most basic application of The Compactness Theorem, as envisaged by Hamilton and realised fully by Perelman, by blowing up a singularity to obtain a limit ancient Ricci flow modelling the singularity. To do this we will take a brief detour to mention Perelman’s “No Local Collapsing Theorem”.

The rough idea of how this looks in practice in 3D will be explained. But to fully illustrate this application, and also some other key ideas – particularly Perelman’s notion of $\kappa$-solutions and their most basic theory – we will focus on the 2D situation, and give a Perelman-style proof of the beautiful results of Hamilton and Chow explaining what Ricci flow does to an arbitrary compact surface.

From there we will consider the problem of starting a Ricci flow with a completely general metric, typically on a noncompact surface. This raises some interesting well-posedness issues as one struggles to find the right way of posing the problem to obtain both existence and uniqueness of solutions. We will resolve this problem with the notion of instantaneously complete Ricci flows.

The lectures will then complete a full circle by applying this 2D theory in order to understand better Hamilton’s compactness theorem and the various extensions that one needs (or desires) to take the theory further. More precisely, we will use the 2D theory (including some additional examples of ‘contracting cusp’ Ricci flows) to construct some visual examples which violate some extended forms of Hamilton’s result that were previously widely believed and used.

Jeff Viaclovsky: Riemannian curvature functionals

We will discuss the basic theory of Riemannian functionals, starting with the total scalar curvature functional, and then concentrating on quadratic curvature functionals. Topics will include calculation of Euler-Lagrange equations, second variation and Jacobi operator, Ebin slice theorem and structure of moduli spaces, and special properties of critical metrics. We will also concentrate on dimension four, and an important class of critical metrics called Bach-flat metrics (which contains the class of anti-self-dual metrics).

Ben Weinkove: The Kahler-Ricci flow on compact Kahler manifolds

In these lectures I will discuss the Kahler-Ricci flow on compact Kahler manifolds and review some results relating the behavior of the flow to the existence of Kahler-Einstein metrics.  The lectures will begin with a brief introduction to Kahler geometry and the Kahler-Ricci flow.  I will then describe some known results on the Kahler-Ricci flow on manifolds with negative, zero and positive first Chern class.  If time permits, I will discuss the behavior of the Kahler-Ricci flow in some examples of manifolds with symmetry.  The lectures will be aimed at graduate students in geometric analysis and no knowledge of Kahler geometry or the Ricci flow will be assumed.

Brian White: Minimal submanifolds

An introduction to minimal submanifolds: basic properties, key curvature estimates and compactness theorems, and existence and regularity of least area surfaces.

Steve Zelditch: Global Harmonic Analysis

Global harmonic analysis refers to the use of global properties of the geodesic flow of a Riemannian manifold (M, g)  to determine the behavior of eigenfunctions of the Laplacian as the eigenvalue tends to infinity. Behavior means their  concentration and oscillation properties.  Eigenfunctions are the quantum states of fixed energy and one would like to know their sizes (measured by $L^p$ norms) or shapes (e.g. their nodal or critical point sets).  Relating classical mechanics (the geodesic flow) and quantum mechanics (the wave equation) is the fundamental problem of semi-classical analysis.

The lectures will go over relatively recent results on $L^p$ norms of eigenfunctions and their relations to the geodesic flow, especially when the flow is  completely integrable or ergodic; on restrictions of eigenfunctions to hyper surfaces; on nodal and critical point sets of eigenfunctions; and on more general solutions of Schrodinger or wave equations. They will also cover Gaussian random combinations of eigenfunctions (random landscapes).

Eigenfunctions of eigenvalue $\lambda^2$ resemble polynomials of degree $\lambda$. Just as real algebraic geometry is more difficult than complex algebraic geometry, so nodal and critical point problems are simplest when (M, g) is real analytic and when the eigenfunctions are analytically continued to the complexification of M.  The lectures will cover a number of recent results on nodal problems in the complex domain.