Titles and Abstracts

Richard Schoen | Eduard Looijenga | Clifford Taubes |
Vaughan F.R.Jones | Cedric Villani | Fernando Marques

Richard Schoen, Stanford University, 2011-9-22/29, 10-13

Title: Highlights in Differential Geometry: 1950 to the present

Abstract

S. S. Chern was a leader of global differential geometry in the 20th century. Since 1950 the subject has seen dramatic developments particularly through the introduction of deep analytic methods. In the first lecture we will highlight some of the main problems of differential geometry. In the second lecture we will survey some of the deep problems which have been solved by variational techniques, and we will briefly describe some of the important geometric variational problems including the Plateau problem and the Yamabe problem. In the third lecture we will introduce geometric flows including the Ricci flow and survey some of the results achieved by flow methods. The first lecture is intended for a general audience with little mathematical training. The second and third should be accessible to a general mathematical audience including advanced undergraduate math majors.

Eduard Looijenga, Tsinghua University, 2011-12-7/14

Title: Invariants and automorphic forms with algebraic geometry as a go-between

Abstract

In the 19th century it was found that a nonsingular cubic plane curve may be viewed as a complex torus and that its basic invariant, the j-invariant, is not only expressible in terms of the coefficients of a defining equation, but also in terms of the periods of the associated elliptic integral. In other words, the moduli space of nonsingular cubic plane curves can be obtained with the help of invariant theory and by means of the theory of automorphic forms. Since then there has been an ongoing conversation between these two areas, often to mutual benefit. But especially the past decade witnessed remarkable progress, leading not only to new and beautiful, higher dimensional examples, but also to substantial developments on a theoretical level.
The lectures will begin with reviewing the basic example and then introduce the audience to some of the work alluded to above.

Clifford Taubes, Harvard University, 2012-5-7/8/10/11

Title: Mysteries of 4 dimensions

Abstract

The first lecture is for a general audience:

Time: 19:00-20:00, May 7 (Monday)

Title: Mysteries of 4 dimensions

Abstract: What with the 3 spatial dimensions and then time, our universe can be viewed as a 4-dimensional space. Astronomers seek to understand the large scale structure of our particular 4-dimensional space. A mathematical question asks for the list of all possible 4-dimensional spaces. I hope to give a sense for what we know and don't know about such a list.

The subsequent three lectures are more technical.

Time: 15:00-16:00, on May 8 (Tuesday), May 10 (Thursday) and May 11 (Friday)

Title: New tools for exploring 3 and 4 dimensions: SL(2;C) connections with bounds on curvature

Abstract: Non-Abelian gauge theories are used to study the structure of 4-dimensional spaces; and all such applications require a theorem of Karen Uhlenbeck about connections with integral bounds on their curvatures. Uhlenbeck's theorem only applies to a gauge theory with compact Lie group. I will describe an extension of Uhlenbeck's theorem that applies to gauge theories with the non-compact group SL(2,C). This extension will likely have analogs for the higher rank non-compact groups; and it may have an analog for certain generalizations of the Seiberg-Witten equations.

Vaughan F.R.Jones, Vanderbilt University, 2011-12-19/21/24/26

Title: Mysteries of 4 dimensions

General lecture: What is a von Neumann algebra?

Abstract: Von Neumann was motivated by many things in his introduction, with Murray, of "Rings of Operators", now called von Neumann algebras: unitary group representations, abstract algebra, projective geometry, operator theory, ergodic theory and most of all, the mathematical structure of quantum physics. I will describe how all these threads are united by the seductively simple notion of a closed *-algebra of operators on Hilbert space. I will also sketch the development of the subject by von Neumann, and beyond, including the notion of subfactor.

Lecture two: Von Neumann algebras and quantum physics.

Abstract: The strong and weak operator topologies have direct physical meaning so that the closedness of a von Neumann algebra is as natural to quantum physics as the completeness of the reals is tp classical physics. Haag and Kastler postulated a model-independent framework for quantum field theory in which loalised observables form von Neumann algebras. This structure has turned out to be extraordinarily rich and has in turn enriched the theory of von Neumann algebras.

Lecture three: Von Neumann algebras and topology.

Abstract: Of the many interactions between von Neumann algebras and topology I will discuss two-the first being the study of manifolds via the group von Neumann algebra of their fundamental group, for instance Atiyah's L^2 index theorem, and second the development of 2+1 dimensional topological quantum field theory beginning with the Jones polynomial which originated in sufbactors.

Lecture four: Von Neumann algebras and random matrices.

Abstract: The large N limit of a single self-adjoint random NxN matrix is particularly well understood even in the presence of a quite arbitrary potential. But for more than one random matrix the theory is less satisfactory. This is because of the non-commutative nature of the beast. The work of Voiculescu shows that even in the purely Gaussian case we land inevitably in the world of type II factors, and indeed we land right on top of some deep unsolved problems in type II factors. I will describe this and more recent work of Guionnet, Shlyakhenko and myself where we create matrix models with a real (non-integer) number of random matrices.

2014 Tsinghua University Shiing-Shen Chern Distinguished Lecture

Nigel Hitchin, University of Oxford, 2014-10-20/24

Lecture 1: Higgs bundles – a survey

Lecture 2: Higgs bundles for diffeomorphism groups

Abstract:

Professor Nigel Hitchin is a leading figure in differential geometry in the past thirty years. He is particularly well known for his originality and creativity. His ability to identify links among different research paradigms open new research directions for many to follow.

Among his numerous research accomplishments, in 1987 and in collaboration with three physicists, he articulated the subject of hyper-Kähler manifolds and its deep relation with super-string theory. Through this piece of work, he lays the mathematical foundation for the research of hyper-Kähler geometry in the past quarter century. It serves as a key impetus to inspire the development of geometry of special holonomy.

Based on his work on Yang-Mills Theory and Higgs' Monopoly Theory, Prof Hitchin discovered what is now known as Hitchin system. One example is the Higgs Bundle. While the development of this theory in its first fifteen years was mainly within the realm of integrable, due to its fundamental nature it has become a key tool in algebraic number theory and representation theory.

In his latest analysis of Higgs bundles, hyper-Kähler geometry emerges again.

Cedric Villani, University of Lyon, 2015-1-6/7

* Generalist lecture: On curvature and optimal transport: When Riemann, Monge and Boltzmann meet

* Specialized lecture: Synthetic theory of Ricci curvature: past, present and tentative future

Abstract:

At the turn of the 21st century, it was realized that Ricci curvature could be captured with a combination of probability theory and optimal transport, thus leading to new advances in differential geometry.

In the first lecture I will recall the fundamentals and overview of this area.

In the second lecture I will review the present state of the resulting synthetic theory of Ricci curvature, including remarkable recent advances made by a number of researchers on "Riemannian" spaces with a weak curvature-dimension bound.

Fernando Marques, Princeton University, 2016-5-10/12

Title: "Min-max theory for the area functional - a panorama"

Abstract: In these lectures we will give a current panorama of the min-max theory for the area functional, initially devised by Almgren in the 1960s and improved by Pitts (1981). This is a deep high-dimensional generalization of the study of closed geodesics. The setting is very general, being that of Geometric Measure Theory, and the main application until very recently was the construction of minimal varieties of any dimension in a compact Riemannian manifold.

In the past few years we have discovered new applications of this old theory, including a proof of the Willmore conjecture, of the Freedman-He-Wang conjecture, and of Yau's conjecture (about the existence of infinitely many minimal hypersurfaces) in the positive Ricci curvature setting. We will give an overview of these results, describe our current efforts to develop the theory further and discuss future directions. Among the most recent developments, we will describe our work on the problems of the Morse index and multiplicity of min-max minimal hypersurfaces. Most of the material covered in these lectures is based on joint work with Andre Neves.