# Seminar (2014)

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

# Seminar (2013)

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

#### Prof. Zhen-Qing Chen (University of Washington, Seattle): 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.

# Seminar (2010)

#### Kenneth Falconer(University of St-Andrews,UK): Symmetry and Enumeration of Fractals. Place: Room 1304, New Science Building, Tsinghua University.

Abstract: I will discuss the construction of certain families of self-similar fractals which display various symmetries. I will then indicate how methods from group theory may be used to analyse the symmetries and enumerate the fractals in such families with each symmetry group. (file ) (movie1 ) (movie2 ) (movie3 ) (movie4 ) (movie5 ) (movie6 )

# Seminar (2009)

#### Alexander Grigoryan(University of Bielefeld, Germany): On positive solutions of semi-linear elliptic inequalities on complete Riemannian manifolds. Place: Room 1304, New Science Building, Tsinghua University.

Abstract: We consider elliptic inequalities of the type $\Delta u+u^{\sigma }\leq 0$ on geodesically complete Riemannian manifolds and provide sharp sufficient conditions in terms of capacities and volumes for the non-existence of positive solutions. This is a joint work with V.A.Kondratiev

#### Jiaxin Hu: 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.

Abstract: This talk is divided into two parts. The first part is for the general audience where I will give a brief introduction of fractal geometry, such as the concepts of self-similar set, Hausdorff measure and Hausdorff dimension. The second part is for more restricted audience where I will survey my recent joint work with Alexander Grigoryan (Bielefeld) and Ka-Sing Lau (Hong Kong) on heat kernels on metric measure spaces, including fractal domains.

# Seminar (2007)

### Place: Room 2203, Math Building

#### 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.

#### 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.

#### 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.

#### 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.