Summer Graduate School

<p>A real-world Flying Wing. Lai proved Hamilton&rsquo;s conjecture on the existence of Flying Wing Ricci solitons.</p>

Ricci flow is the nonlinear heat-type equation for Riemannian metrics given by ∂tg = −2Ric. It is a powerful tool in the intersection of geometry, analysis, and topology, and has been used by Perelman and Hamilton to resolve both the Poincar´e conjecture and the more general Thurston geometrization conjecture. The approach is to start with any Riemannian metric g0 on any closed 3-manifold, and run the Ricci flow with g0 as the initial metric. As a heat equation, it smooths out metrics. However, since it is nonlinear, singularities can and usually will develop. The method to approach the geometrization conjecture is to devise a geometric and topological surgery procedure, which in turn is based on understanding singularity formation under Ricci flow. Ricci flow was successful in dimension 3 largely due to the classification of singularity models in this dimension. Furthermore, in dimension 3, singularity models turn out to be exactly the same as Ricci solitons, which are the self-similar solutions to Ricci flow. Thus, Ricci solitons have come to the forefront in the theory of Ricci flow. In dimension 4, although the theory of Ricci solitons is well-developed, there are many important remaining questions. In particular, there is a substantial qualitative theory of the curvature, volume, and number of topological ends of Ricci solitons. However, the current knowledge is far short of a classification.

Participants will learn the basic theory of Ricci flow and Ricci solitons—they will learn the fundamental equations and inequalities, how to do basic tensor calculations, how to control the potential function, how to obtain global curvature bounds, how to do bootstrap arguments, how to control the geometry at infinity, how to construct new Ricci solitons, the current cutting edge existence theory for Ricci solitons, including the existence and classification of flying wing solitons. Many arguments involve the maximum principle, integration by parts, and/or bootstrap arguments. In short, the student will learn the art of the a priori estimate. This is one of the main methods for understanding the geometry of Ricci solitons.

School Structure

There will be 20 total lectures (2 per day). The lectures are organized into two parallel tracks: a Foundational Track (focusing on the machinery of Ricci flow) and an Advanced Track (focusing on the analysis and construction of Solitons). Each lecture will be followed by a problem and collaborative session.  These sessions will be led by the teaching assistants in coordination with the relevant lecturer for that day.

Prerequisites

The participant should be familiar with smooth manifolds, tensor bundles, the Levi-Civita connection, the Riemann curvature tensor (and its traces), Bianchi identities, and basic comparison theory. Prior knowledge of geometric flows is not required; we will build the theory from the ground up.

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.