Summer School on Perspectives in Geometric Analysis
Speaker:
Sponsored by:
Time: From 2016-06-26 11:37 To 2016-06-30 11:37
Venue: Lecture Hall, Jiayibing Building, BICMR
Speakers List:
RICHARD H. BAMLER Berkeley
Xianzhe
Dai UCSB
Yannick Sire Johns Hopkins University
Jeff
Streets UC Irvine
Rivière Tristan ETH Zurich
Xiaodong Wang Michigan State University
Titles& Abstracts:
RICHARD H. BAMLER (Berkeley)
Title: Ricci flows with bounded scalar curvature
Abstract:
We discuss recent results on the structure theory of Ricci flows with bounded scalar curvature. Such flows naturally occur in the study of Kaehler Ricci flows on Fano manifolds. The Hamilton-Tian Conjecture states that Kaehler Ricci flows on Fano manifolds converge, away from a subset of codimension larger or equal to 4. The structure theory presented in this mini course is purely Riemannian and it gives an affirmative answer to a (much more general) Riemannian version of the Hamilton-Tian Conjecture.
Xianzhe Dai (UCSB)
Title: Index theory and its geometric applications
Abstract:
The Atyiah-Singer index theory is one of the landmark results in 20th century mathematics, unifying several great theorems in differential geometry, algebraic geometry, and differential topology. In the first lecture, we will review the Dirac operator, characteristic classes, the Atiyah-Singer index theorem, as well as the Atiyah-Patodi-Singer index theorem. The remaining two lectures will concentrate on the geometric applications. The second lecture will concern the stability issues of Einstein metrics, and the third one the construction of interesting geometric invariants which can distinguish various geometric structures.
Yannik Sire (Johns Hopkins University)
Title: Geometric analysis of solutions to semi linear equations: flatness and convexity of level sets with their links to minimal surfaces
Abstract:
This mini-course is devoted to the study of geometric properties of solutions to semi linear equations. I will first describe a well known conjecture by De Giorgi, explaining the link with minimal surfaces. Then I will move on two types of generalizations: the case of fractional order operators and the case of Riemannian manifolds. Finally, I will describe a new counter-example of convexity for solutions of semi linear equations in convex domains.
The common theme among these topics is to investigate the geometry of level sets of solutions.
Jeff Streets (UC Irvine)
Title: Generalized Kahler-Ricci flow
Abstract:
In this minicourse I will introduce geometric and analytic aspects of the generalized Kahler-Ricci flow. Generalized Kahler geometry was originally discovered through investigations into supersymmetry, and was later rediscovered in a purely geometric context in work of Hitchin/Gualtieri. The generalized Kahler-Ricci flow was introduced in joint work of myself and G. Tian and is a natural tool for addressing questions of existence and moduli of canonical metrics and the topology of generalized Kahler manifolds. We will begin with a brief introduction to generalized Kahler geometry, and then introduce fundamental aspects of the equation. In the final lecture we will prove global existence and convergence results, and what open problems remain.
Rivière Tristan (ETH Zurich)
Title
:The Variations of Yang-Mills Lagrangian
Abstract
:
Yang-Mills theory is growing at the interface between high energy physics and mathematics.
It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential topology, Hitchin Kobayashi conjecture relating holomorphic bundles over Kähler manifolds and Mumford stability in complex geometry or also Gromov Witten invariants in symplectic geometry...etc. While the influence of Gauge theory in geometry is quite notorious, one tends sometimes to forget that Yang-Mills theory has been also at the origin of fundamental progresses in the non-linear analysis of Partial Differential Equations in the last decades.
The purpose of this mini-course is to present the variations of this important lagrangian. We shall raise analysis question such as existence and regularity of Yang-Mills minimizers or critical point of Yang-mills lagrangian in general.
We will first describe during the first half of the course the progresses which have been made in these directions for the critical dimension 4 and below. At this occasion we will mostly insist on the important contributions by K.Uhlenbeck from the late seventies - early eighties. The second part of the mini-course will be devoted to the study of Yang-Mills fields in dimension larger than 4. We will first describe the Lin-Tian concentration compactness method in super critical dimension as well as the epsilon-regularity results obtained by Tao-Tian and Meyer-Rivière.
In the last part of the course we shall present very recent results obtained in collaboration with M. Petrache regarding the ad-hoc framework for producing Yang-Mills critical points in dimension larger than 4.
Xiaodong Wang (Michigan State University)
Title: Comparison and rigidity results on compact Riemannian manifolds with boundary
Abstract:
For compact Riemannian manifolds with nonempty boundary, it is interesting to study the relationship between the geometry on the boundary and geometry of the interior. I will discuss comparison and rigidity results for manifolds with a lower bound for the Ricci curvature. The focus will be on sharp geometric inequalities that yield rigidity results in the equality case.
If time allows I will also discuss results for manifolds with a lower bound for the scalar curvature. Many results and questions in this direction are motivated by general relativity.