Heat kernel estimates and Harnack inequalities for symmetric non-local Dirichlet forms and their applications.

( slides)

Place: Room A-404, New Science Building, Tsinghua University.

On semi-linear elliptic inequalities on Riemannian manifolds.

Place: Room A-404, New Science Building, Tsinghua University.

Heat kernels of some Schroedinger operators. ( slides)

Place: Room A-304, New Science Building, Tsinghua University.

Homologies of digraphs and Kuenneth formula. ( slides)

Place: Room A-404, New Science Building, Tsinghua University.

Homology and homotopy of finite graphs.

Place: Room A-304, New Science Building, Tsinghua University.

Abstract: We introduce the notions of homology and homotopy for digraphs (directed graphs) and present their basic properties. In particular, homology groups are homotopy invariant, and the first homology group is abelization of the fundamental group.

On nonnegative solutions for semilinear elliptic inequalities on complete Riemannian manifolds.

Place: Room A-304, New Science Building, Tsinghua University.

Abstract: We provide optimal condition in terms of the volume growth of a Riemannian manifold that ensures that any non-negative solution to the inequality $\Delta u + u^{\sigma}\le 0$ on this manifold is identically equal to 0. ( pdf)

Brownian Motion on Spaces with Varying Dimension.

Place: Room A-404, New Science Building, Tsinghua University.

Abstract: Brownian motion is a building block of modern probability theory.It has important and intrinsic connections to analysis and partial differential equations.In real world, there are many examples of spaces with varying dimensions. For example, image an insect moves randomly in a plane with an infinite pole installed on it.In this talk,I will introduce and discuss Brownian motion on a state space with varying dimension, as well as its infinitesimal generator. I will present sharp two-sided estimates on its transition density function (also called heat kernel).The two-sided estimates is of Guassian type but the parabolic Harnack inequality fails for such process and the measure on the underlying state space does not satisfy volume doubling property.

Approximation of fractals by tubular neighborhoods - geometric and analytic properties.

Place: Room A-304, New Science Building, Tsinghua University.

Heat kernels and Green functions on metric measure spaces.

Place: Room A-203, New Science Building, Tsinghua University.

Heat kernel estimates of non-local Dirichlet forms on metric spaces.

Place: Room 2203, New Science Building, Tsinghua University.

Symmetry and Enumeration of Fractals.

Place: Room 1304, New Science Building, Tsinghua University.

Random walks and analysis on graphs (III).

Place: The Morningside Center, Chinese Academy of Sciences (CAS).

Random walks and analysis on graphs (II).

Place: The Morningside Center, CAS.

On positive solutions of semi-linear elliptic inequalities on complete Riemannian manifolds.

Place: Room 1304, New Science Building, Tsinghua University.

Random walks and analysis on graphs (I).

Place: The Morningside Center, CAS.

分形世界---一个破碎的梦: 从康托 集和他的忧郁症说起(in memory of Loo-Keng Hua (华罗庚) on the occasion of his 99th birthday).

Place: 郑裕彤讲堂, 清华大学.

#### March 15, 2007.

#### Jacques Peyriere (Universite de Paris XI).

#### March 29, 2007.

#### Thomas Duquesne (Universite de Paris XI):

Geometric and fractal aspects of Levy trees.#### April 5, 2007.

#### Julien Barral (INRIA, Paris):

On the multifractal analysis of discontinuous measures.Abstract: We will explain how the multifractal formalism imposes to seek for classes of discontinuous measures for which the multifractal formalism holds. Then we will show several ways to construct such measures, and we will focus in particular on a class which is closely related to the continuous Mandelbrot multifractal measures and in some sense unifies stable Levy subordinators and Mandelbrot measures. Finally, we will explain that the multifractal analysis of these measures requires a new theorem on ubiquitous systems. The results that will be presented were obtained in joint works with S. Seuret.

#### April 12, 2007.

#### Patrice Le Calvez (Universite de Paris XIII ):

From Brouwer translation theorem to the studies of homeomorphisms of surfaces.Abstract: We will state an equivariant foliated version of the Brouwer's Plane Translation Theorem that implies existence of dynamically transverse foliations for homeomorphisms of surfaces that are isotopic to the identity. Many applications can be obtained: we will explain why any time one map of an Hamiltonian isotopy on a surface of genus greater than 1 has at least three contractible fixed points and infinitely many contractible periodic points.

#### April 19, 2007.

#### Yong Lin (People's University, Beijing):

Ricci curvature on locally finite graphs.Abstract: We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that this is always can be done on locally finite graphs. We also get a estimate for the eigenvalue of Laplace operator on finite graphs.

#### April 26, 2007.

#### Ka-Sing Lau(Chinese University of Hong Kong):

On some aspects of self-affine fractals.Abstract: Self-affine sets are the attractors of iterated function systems of affine maps. This class of sets plays a central role in fractals geometry and the theory is in an exciting stage of development. It is a topic in the cross road of the areas of analysis, probability, ergodic theory and number theory. The talk is expository in nature, we will discuss the on-going work on the tiling problems, the spectral set problems and the related projects.

#### Dec 13, 2007.

#### Alexander Grigoryan(University of Bielefeld, Germany):

Heat kernels of Schrodinger operators and stability of the Harnack inequality.

Place: The Morningside Center, CAS.#### Dec 15, 2007.

#### Alexander Grigoryan(University of Bielefeld, Germany):

Heat kernels on manifolds with ends.

Place: The Morningside Center, CAS.#### Dec 22, 2007.

#### Alexander Grigoryan(University of Bielefeld, Germany):

Heat kernels on metric measure spaces.

Place: The Morningside Center, CAS.