# Oberseminar Geometric Analysis

SFB 701, project A6

# Sommersemester 2010

20.04.10  10:15   U2-135
Wolfhard Hansen (Bielefeld)
3G-inequality for planar domains

27.04.10  10:15   U2-135
Markus Biegert (Ulm)
Does diffusion determine the manifold?

04.05.10  10:15   U2-135
Xueping Huang  (Bielefeld)
Stochastic completeness of graphs

14.05.10  14:15   V4-116
Spherically symmetric graphs

 Abstract: We introduce the class of spherically symmetric graphs of unbounded geometry.  We will then discuss their stochastic completeness and basic spectral properties.  We also show how to use this class to obtain some comparison results.

18.05.10  10:15   U2-135
Jun Masamune (Pennsylvania State University, USA)
Homogenization and its application to probability theory

 Abstract: I will show the Mosco convergence of the Dirichlet forms which is defined on a cylinder with highly conductive boundary. The tightness of the associated Wiener measures will be shown, and as a conclusion, we will obtain the weak-convergence of those Wiener measures.

25.05.10  10:15   U2-135
Bartlomiej Dyda (Bielefeld)
On inequalities concerning functions of fractional smoothness

01.06.10  10:15   U2-135
Shun-Xiang Ouyang (Bielefeld)
Heat kernel estimates and  functional inequalities on Dirichlet spaces

08.06.10  10:15   U2-135
Alexander Bendikov (University of Wroclaw / Bielefeld)
On a class of random walks on ultra-metric spaces 1

15.06.10  10:15   U2-135
Alexander Bendikov (University of Wroclaw / Bielefeld)
On a class of random walks on ultra-metric spaces 2

22.06.10  10:15   U2-135
Jun  Masamune (Pennsylvania State University, USA)
The conservation property of non-local Dirichlet forms

 Abstract: In this talk, I would like to show the conservation property of a non-local regular Dirichlet form on an Euclidean space with Radon measure under a volume growth condition. Our approach is to develop the integral deviation property and to apply the method by M.P. Gaffney. If the time permits, I would like to talk also on the L^p Liouville property for the same forms. Both results were obtained in joint works with Toshihiro Uemura.

29.06.10  10:15   U2-135
Igor Verbitsky (University of Missouri, USA)
Nonlinear equations with natural growth terms

06.07.10  10:15   U2-135
Jiaxin Hu (Tsinghua University, Beijing)
Generalized Bessel and Riesz potentials on metric measure spaces

13.07.10  10:15   U2-135
Alexander Loboda (Voronezh, Russia)
Homogeneity of embedded  manifolds

 Abstract: We discuss the problem of description of real hypersurfaces that are (locally) homogeneous with respect to different structures (holomorphic, affine, projective) of the ambient complex spaces of small dimensions. The history of this problem goes back to 1932 when E. Cartan gave a complete solution of the holomorphic homogeneity problem in the case of 2-dimensional complex spaces. It was not until 1995 when a comprehensive description of affinely homogeneous surfaces of 3-dimensional real space was obtained by Doubrov, Komrakov and Rabinovich.  We describe a coefficient approach to the problem, that is connected to the use of the canonical equations of the manifolds under consideration and to the description of Lie algebras of vector fields on these surfaces. Recent results of the author  about the homogeneity in 2- and 3-dimensional complex spaces will be presented.

20.07.10  10:15   U2-135
Satoshi Ishiwata (Tsukuba University, Japan)
Heat kernel estimates on non-compact gluings

# Wintersemester 2009/2010

09.09.09  10:15   V3-201
Jiaxin Hu (Tsinghua University, Beijing)
Comparison inequalities for heat kernels

10.09.09  10:15   V3-201
Alexander Bendikov (University of Wroclaw)
Spectral properties of some random walks on locally finite groups

11.09.09  10:15   V3-201
Boguslaw Zegarlinski  (CNRS Toulouse)
Ergodicity of Markov semigroups in infinite dimensions

14.09.09  10:15   V3-201
Naotaka  Kajino (Kyoto University, Japan)
Short time asymptotics of the heat kernels on the usual and harmonic Sierpinski gaskets

15.09.09  10:15   V3-201
Satoshi  Ishiwata  (Tsukuba University, Japan)
Smallness of the first Neumann eigenvalues under a bottleneck heat kernel estimate

16.09.09  10:15   V3-201
Wolfhard  Hansen (Bielefeld University)
One radius results for supermedian functions on Rd, d ≤2
17.09.09  10:15   V3-201
Ivan  Netuka (University of Prague)
An excursion into infinite dimensional convexity

21.09.09  10:15   V3-201
Minoru Murata (Tokyo Institute of Technology)
Structure of non-negative solutions to parabolic equations and perturbation theory for elliptic operators

22.09.09  10:15   V3-201
Wolfhard Hansen (Bielefeld University)
Harnack inequality and Hölder continuity for harmonic functions

23.09.09  10:15   V3-201
Shun-Xiang Ouyang  (Bielefeld University)
Harnack inequalities and applications for Ornstein-Uhlenbeck semigroups with jump

02.10.09  10:15   V3-201
Michiel  van den Berg  (University of Bristol)
Minimization of Dirichlet eigenvalues with geometric constraints

27.10.09  10:15   V3-201
Alexander Teplyaev  (University of Connecticut)
Uniqueness of Brownian motion on Sierpinski carpets, part I

03.11.09  10:15   V3-201
Alexander Teplyaev  (University of Connecticut)
Uniqueness of Brownian motion on Sierpinski carpets, part II

10.11.09  10:15   V3-201
Alexander Teplyaev  (University of Connecticut)
Uniqueness of Brownian motion on Sierpinski carpets, part III

22.12.09  10:15   V3-201
Wei Liu  (Bielefeld)
Dimension-free Harnack inequality and its applications

12.01.10  10:15   V3-201
Nikolai Nikolov (Sofia)
Estimates of invariant metrics on C-convex domains

19.01.10  10:15   V3-201
Uta Freiberg  (Jena/Siegen)
Einstein relation on fractals

02.02.10  10:15   V3-201
Moritz Kaßmann  (Bielefeld)
Jump processes, integro-differential operators, and regularity

# Sommersemester 2009

30.04.09  12:00 C2-144
Hendrik Vogt (TU Dresden)
Kato class and Gaussian bounds for the heat equation on the half space

07.05.09  12:00 C2-144
Satoshi Ishiwata (Tsukuba University, Japan)
The gradient heat kernel estimate on gluings

14.05.09  14:00 U2-241
Elmar Teufl (Bielefeld)
A determinant identity of Sylvester, an inverse problem concerning effective resistances, and computing the number of spanning trees via electrical network theory

 Abstract: Given an electrical network (a finite set of nodes and resistances between pairs of nodes), there is, for every pair of nodes, a number called effective resistance, so that the network with respect to this pair of nodes is equivalent to a single wire equipped with the effective resistance. Given the \$\binom{n}{2}\$ resistances we can compute the set of \$\binom{n}{2}\$ effective resistances (e.g. using Schur complements). We are interested in the inverse problem. Given effective resistances, can we recover the original resistances? Kigami showed that there is at most one network solving this inverse problem without giving an explicit solution. We will give an explicit solution using an old determinant identity due to Sylvester and point out a connection to the problem of counting spanning trees.

18.06.09  16:00 C2-144
Wolfhard Hansen (Bielefeld)
Harmonic measures for a point may form a square

25.06.09  16:00 C2-144
Igor Verbitsky (Missouri)
Quasilinear and Hessian equations and inequalities

02.07.09  16:00 C2-144
Gerasim Kokarev (University of Edinburgh)
Variational aspects of Laplace eigenvalues on Riemannian surfaces

23.07.09  16:00 C2-144
Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains

# Sommersemester 2008

23.06.08  16:00 S2-121
Wolfhard Hansen (Bielefeld)
Density of extremal measures in parabolic potential theory

16.06.08  16:00 V3-201
Verfahren W2 Analysis

09.06.08  16:00 V3-201
Verfahren W2 Analysis

02.06.08  16:00 S2-121
Denis Labutin (Bielefeld / UCSB)
Partial regularity for the Monge-Ampere equation

26.05.08  16:00 V3-201
Verfahren W2 Analysis

19.05.08  16:00  S2-121
Elton P. Hsu  (Northwestern University)
Bismut's formula for heat kernels on vector bundles

05.05.08  16:00  S2-121
Dan Mangoubi  (MPIM Bonn)
Geometry of nodal domains

# Wintersemester 2007/8

31.01.08  16:00  V2-216
Denis Labutin (Bielefeld / UCSB)
Critical regularity for non-linear elliptic equations

24.01.08  16:00  V2-216
Moritz Kassmann (Bonn)
Critical regularity questions for local and non-local problems

17.01.08  16:00  V2-216
Laurent Saloff-Coste (Cornell)
Analysis on inner uniform domains

06.12.07  16:00  V2-216
Jun Kigami (Kyoto)
Analysis on fractals

29.11.07  16:00  V2-216
Alexander Grigoryan (Bielefeld)
Heat kernels on metric measure spaces
 Abstract: The recent development of Analysis on fractal spaces has motivated the study of abstract heat kernels on metric measure spaces. An abstract heat kernel is a function on such a space that by definition satisfies certain standard properties of the classical heat kernels, such as positivity, symmetry, the semi-group property, approximation of identity, and stochastic completeness. Such a heat kernel is associated with a Markov process on the space in question and is, in fact, its transition density. In many cases of interest, heat kernels satisfy two-sided estimates of two types: (1) Gaussian or, more generally, sub-Gaussian estimate, which is typical for diffusion processes; (2) estimate with a polynomial tail, which is typical for symmetric stable processes. One of the main results is that, under certain mild assumptions, these two cases exhaust all possible heat kernel estimates of homogeneous type.

22.11.07  V2-216
16:00-17:00
Ben Hambly (Oxford)
Local limit theorems for random walks on percolation clusters
 Abstract: The talk will consider the simple random walk on the supercritical clusters of bond percolation in the d-dimensional integer lattice. It is now well known that there are Gaussian heat kernel estimates and a quenched invariance principle for this process. We establish a parabolic Harnack inequality and show that this leads to a local limit theorem for the convergence of the transition kernel for the random walk to the Gaussian heat kernel. These ideas can be extended to other random walks on sequences of graphs.
17:00-18:00
Christophe Sabot (Lyon)
Limit laws for one dimensional random walks in random environments
 Abstract: In 1975, Kesten, Kozlov and Spitzer proved a limit theorem for one dimensional random walks in random environments with null speed. They proved that the RWRE properly renormalised converges to a stable law, but they left open the explicit description of the parameters of this law. In this talk, based on a joint work with N. Enriquez and O. Zindy, I will present a different proof of this limit theorem, based on the analysis of the potential associated with the environment, which leads to a description of the parameters of the law. The case of beta environments appears to be particularly explicit.

15.11.07  16:00  V2-216
Asymptotic behaviour  of solutions of nonlinear diffusion equations

08.11.07  16:00  V2-216
Wolfhard Hansen  (Bielefeld)
Convexity properties of harmonic measures

25.10.07  16:00  V2-216
Alexander Bendikov (Wroclaw/Bielefeld)
Long time decay of the return probability for random walks without second moment

18.10.07  16:00  V2-216
Christoph Richard (Bielefeld)
Random colourings of aperiodic graphs: Ergodic and spectral properties.

04.10.07  16:00  V3-204
Takashi Kumagai (Kyoto)
Heat kernel estimates and Harnack inequalities for jump processes
 Abstract: I will talk about the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. This is a joint work with M.T. Barlow, A. Grigor'yan and with M.T. Barlow, R.F. Bass.

02.08.07  V2-216
14:00-15:00  Wolf-Jürgen Beyn (Bielefeld)
Nonlinear stability of patterns and the method of freezing

15:30-16:30  Sergey Bobkov (Minnesota)
Poincaré-type inequalities for multidimensional convex bodies and log-concave probability distributions

# Sommersemester 2007

Di  14:30-17:00    T2-213

17.04.07  Jiaxin Hu (Beijing)
Domains of Dirichlet forms and the effective resistance estimates on p.c.f. fractals

08.05.07  Gilles Carron (Nantes)
Riesz transform and connected sum

 Abstract: We will present recent  results (both positive and negative) on the following question. Assume that M is a complete Riemannian manifold such that the Riesz transform D-1/2 is bounded on Lp for a certain p in ]1,∞[. Is it true that the Riesz transform is bounded on the Lp on the connected sum M#M?  This is partially a joint work with T. Coulhon (Cergy) and A. Hassell (Canberra).

05.06.07
14:30-15:30  Michiel Van den Berg (Bristol)
Heat flow and Hardy inequality in complete Riemannian manifolds with singular initial conditions

16:00-17:00  Yuri Kondratiev (Bielefeld)
Jump generators related to some spatial Markov processes

19.06.07  June Masamune (Worcester, USA)
Conservative principle for differential forms

26.06.07   14:00-16:00
Guido Elsner (Bielefeld)
Geometric and probabilistic estimates of higher-order eigenvalues of finite Markov chains

03.07.07   Alexander Teplyaev (Connecticut, USA)
Spectral analysis and Gaussian type heat kernel estimates on the Sierpinski gasket type fractals
10.07.07  Ivan Netuka (Prague)
Potential theory of the farthest point distance function

# Wintersemester 2006-7

Mi  15:30-18:00    U5-133

08.11.06  Andras Telcs (Budapest)
Heat kernels on graphs, isoperimetric inequalities, and Einstein relation

15.11.06  Wolfhard Hansen (Bielefeld)
Convexity of limits of harmonic measures

22.11.06  Laurent Saloff-Coste (Cornell)
Heat kernels, Neumann and Dirichlet, in intrinsically uniform domains
 Abstract: There are many great work dealing with the Dirichlet problem and the Martin boundary of Euclidean domains, starting with the paper by Hunt and Wheeden (1970) on the Martin Boundary of Lipschitz domains, the papers of Dahlberg (1977), Ancona (1978) and Wu (1978), the work of Jerison and Kenig (1982) and more recent work of Aikawa. These works are concerned with the elliptic theory. Aspects of the parabolic case appears in work  by E. Fabes and M. Safonov and their collaborators including Garofalo, Salsa, and Yuan. Strangely enough, the heat kernel is absent of most (if not all) of these works and only appeared quite recently in work of N. Varopoulos (and also Q.S. Zhang and R. Song). In this talk, I will discuss a recent joint work with my graduate student Pavel Gyrya where we study the heat kernel (with either Neumann or Dirichlet boundary conditions) in a large class of domains that includes the domain above the graph of any Lipschitz function and the exterior of any convex set in Euclidean n-space.

29.11.06  jointly with the probability seminar, room V3-201
Anatoly Vershik (St. Petersburg)
Classification of the metric spaces with measure and random matrices
 Abstract: There is a simple classification of the metric triples (=metric spaces with measure) up to measure preserving isometries  (Gromov-V), which is a particular case of the classification of the measurable functions of several variables. The invariant is a measure on the space of so called distance matrices. There are many open problems related to this subject.

06.12.06  Elmar Teufl (Bielefeld)
Simple random walks on nice trees

17.01.07  Alexander Grigoryan (Bielefeld)
Function theory on metric measure spaces and application to stochastic processes

24.01.07
15:30-16:30 Andras Telcs (Budapest)
Potential theory on graphs

17:00-18:00 Sergey Piskarev (Moscow)
Crank-Nicolson scheme for abstract linear systems

31.01.07  jointly with the probability seminar, room V3-201
Gerhard Huisken (Potsdam)
Isoperimetric inequalities via curvature flows
 Abstract: Geometric evolution equations for hypersurfaces can be used to prove several sharp isoperimetric inequalities. The lecture concentrates on the analytical aspects of this method

07.02.07  jointly with the probability seminar, room V3-201
Theo Sturm (Bonn)
Optimal transportation, Ricci curvature and diffusions on the L2-Wasserstein space
 Abstract: We introduce and analyze generalized Ricci curvature bounds for metric measure spaces (M;d;m), based on convexity properties of the relative entropy Ent(:jm). For Riemannian manifolds, Curv(M;d;m) ¸ K if and only if Ric(M) ¸ K; for the Wiener space, Curv(M;d;m) = 1. One of the main results is that these lower curvature bounds are stable under (e.g. measured Gromov-Hausdorff) convergence. This solves one of the basic problems in this field, open for many years. Furthermore, we introduce a (more restrictive) curvature-dimension condition CD(K;N) which implies sharp versions of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows to construct a canonical Dirichlet form with Gaussian bounds for the corresponding heat kernel. Finally, we indicate how to construct a canonical reversible process on the L2-Wasserstein space of probability measures P(R), regarded as an infinite dimensional Riemannian manifold. This process has an invariant measure P¯ which may be characterized as the 'uniform distribution' on P(R) with weight function 1/Z exp(¡¯ ¢Ent(:jm)) where m denotes a given finite measure on R. One of the key results is the quasi-invariance of this measure P¯ under push forwards ¹ 7! h¤¹ by means of smooth di®eomorphisms h of R.