|Center of Mathematical Sciences, Zhejiang University|
Perspectives and open problems in geometric analysis, I.
Titles and Abstracts
Lect 1. The definition of the eigenvalue problems and their significance; gradient estimates of eigenfunctions & Li-Yau's method on the lower bound of the first eigenvalue.
Lect 2. The gap of the first two eigenvalues. We will prove that the gap estimate is a sub-problem of the problem on the first Neumann eigenvalue. This seems to be new.? We will emphasis on the following three special topics: the proof of the Brascamp-Lieb theorem using the continuation method of Singer-Wong-Yau-Yau and the heat flow method; the set up of the gap estimate in terms of the Barkey-Emery Ricci tenor; the set up of the gap estimate in terms of the first Neumann eigenvalue of a domain of higher dimension. Using these settings, we are able to recover all gap estimates by using the estimates for the first Neumann eigenvalue.
Lect 3. The definition of the essential spectrum on a complete non-compact manifold. The variational principle. The $L^p$ spectrum. The theorem of Sturm that the $L^p$ spectrum are the same for all $p$ for Ricci non-negative manifolds.
Lect 4. The theorem of Jiaping Wang on the computation of the essential $L^2$ spectrum for Ricci non-negative complete noncompact manifolds. Wang's theorem, in terms of the method, is parallel to the theorem that any Ricci non-negative manifolds are of at least linear volume growth. Possible generalization of Wang's theorem.
Title: Monotonocity and holomorphic functions I, II, III
Abstract: In these series of lectures I shall explain how one can derive some fundamental monotonicities out of the sharp estimates of Li-Yau-Hamilton type and how they can be useful in the resolution of two conjectures of Yau.
Title: Sobolev Type Inequalities on Complete
Title: Function theory and applications in geometry
Abstract: We will give a partial overview of the study of harmonic functions on complete manifolds and its applications. The topics to be covered will include the stability of the space of the harmonic functions under compact perturbation, the dimension counting of the space of polynomial growth harmonic functions, the splitting theorem of manifolds with maximal bottom spectrum.
Abstract: The spectral theory (eigenvalues, eigenfunctions) of the Laplacian is quantum mechanics on a Riemannian manifold. The associated classical mechanics is the dynamics of the geodesic flow. A fundamental question is to relate the classical and quantum mechanics in the limit of large eigenvalues. When the geodesic flow is chaotic (i.e. ergodic or mixing), the question of how the eigenfunctions reflect the chaotic classical dynamics is called quantum chaos. My three lectures will introduce this topic and discuss some recent results.
Lecture 1: Ergodicity of eigenfunctions.
I will explain how the ergodicity of the geodesic flow implies an equidistribution theorem for eigenfunctions.
Lecture 2: Ergodicity and nodal hypersurfaces
Almost 30 years ago, S.T. Yau conjectured that the nodal hypersurfaces (zero sets) of eigenfunctions satisfied certain volume bounds in terms of the eigenvalue. Several
This sounds impossible. But it turns out to be possible if we analytically continue the eigenfunctions to the complexification of the manifold and study the complex zero
Lecture 3: Nodal lines which touch the boundary of an analytic domain.
In this lecture I explain how to use layer potential theory to reduce some nodal problems on eigenfunctions to the boundary of real analytic domain. I will discuss known results and open problems.
Title: Transversal index and geometric quantizattion on noncompact symplectic manifolds
Abstract: We describe a recent joint work?with Xiaonan Ma where we give?a resolution to Vergne's conjecture on the geometric quantization formula for? compact Lie group actions on?noncompact symplectic manifolds with proper moment maps.
Mark S. Ashbaugh
Title: Open Problems for the Eigenvalues of the Laplacian and Related Operators
Lecture 1: Isoperimetric Inequalities: Survey and Open Problems
We will survey the known isoperimetric inequalities for the
eigenvalues of the Laplacian and related operators and go on to
discuss a number of related conjectures and open problems. In
particular, we will look at low eigenvalues for domains of fixed area
(volume) and at eigenvalue ratios. A discussion of the Polya-Szego
Lecture 2: Universal Inequalities: Survey and Open Problems
Beginning with work of Payne, Polya, and Weinberger in the mid-50's, a variety of universal inequalities for eigenvalues have been established. Later developments include work of Hile and Protter and H. C. Yang. Indeed, the work of Yang has led to many recent developments in the area, especially in the last 10 years or so. We will discuss some of these results and what more might be true.
Lecture 3: Other Interesting Inequalities and Open Problems
This lecture will include discussions of Polya's conjectures for the eigenvalues of the Laplacian (with either Dirichlet or Neumann
boundary conditions) and the related Friedlander inequalities and go
on to address inradius inequalities, the gap conjecture, and
Title: Heat kernels on manifolds with ends
Abstract. Let M be a Riemannian manifold that is a connected sum of k manifolds M_1,...,M_k, each of the being geodesically complete and with non-negative Ricci curvature. The question to be discussed is how to estimate the heat kernel on M. The answer happens to be rather non-trivial? (for example, it depends on whether the manifolds M_i, or some of them, are parabolic or not), and the proof is long and uses a number of results of their own interest. I'll give an account of the proof going through the following steps:
1). The general strategy and gluing techniques. Estimating the heat kernel using the hitting probabilities and the Dirichlet heat kernels in domains.
This is a joint work with L.Saloff-Coste.
Title: Some open problems related with k-Yamabe problem
Abstract: In this talk, I will give a partial overview of the study of k-Yambe problem. The existence, compactness and admissibilty of the solutions to the k-Yamabe equation will be discussed.
For more information,please contact Prof.Weiming Sheng:email@example.com
Address: Mailbox 1511, Center of Mathematical Sciences