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Oberseminar Geometric Analysis

SFB 1283, project A3
IRTG 2235, area D

Sommersemester 2022

The seminar meets in this semester online via Zoom. Time is given in Central European Time Zone.

Friday 25.03.22  7:00-8:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (4)

 

Tuesday 05.04.22  9:00-10:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (5)

 

Tuesday 12.04.22  9:00-10:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (6)

 

Tuesday 19.04.22  9:00-10:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (7)

 

Tuesday 26.04.22  9:00-10:30   
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (8)

 

Tuesday 10.05.22  9:00-10:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (9)

 

Tuesday 17.05.22  9:00-10:30  
 
Alexander Grigor’yan  (Bielefeld)

Overview of path homology theory of digraphs (10)

 

Tuesday 24.05.22  9:00-10:30  
 
Phil Kamtue (Tsinghua, Beijing)

Bakry-Emery curvature on graphs as an eigenvalue problem

 

Abstract:

Bakry-Emery curvature is a notion of Ricci-type curvature (or more precisely, lower Ricci curvature bound) motivated from Bochner's formula in Riemannian geometry. It has been introduced and developed in the setting of weighted graphs by Elworthy (1989), Schmuckenschlager (1996) and Lin-Yau (2010).

 

Here, we propose the method of computing the Bakry-Emery curvature: in short, this curvature is the smallest eigenvalue of a symmetric matrix, which we called "curvature matrix". We then use this formulation to analyze Bakry-Emery curvature as a function of the dimension parameter. As an application, we could simply derive the curvature of Cartesian products.

 

This talk is based on a joint work with David Cushing (Newcastle), Shiping Liu (USTC) and Norbert Peyerimhoff (Durham).

 

 

 

 

Wintersemester 2021/22

The seminar meets in this semester online via Zoom. Time is given in Central European Time Zone.

Tuesday 16.11.21  10:15-11:45  
 
Alexander Tyulenev  (Steklov Institute, Moscow)
Almost sharp descriptions of traces of Sobolev Wp1(Rn)-spaces to arbitrary compact subsets of Rn.
The case p(1,n].

 

Tuesday 23.11.21  10:15-11:45  
 
Alexander Tyulenev  (Steklov Institute, Moscow)
Almost sharp descriptions of traces of Sobolev Wp1(Rn)-spaces to arbitrary compact subsets of Rn.
The case p(1,n]. Part II.

 

Tuesday 30.11.21  9:15-10:45  
 
Jiaxin Hu  (Tsinghua University)
Tail estimates of heat kernels on doubling spaces.

I. Parabolic mean value inequality and on-diagonal upper estimate of heat kernel

 

Tuesday 07.12.21  9:15-10:45  
 
Eryan Hu  (Tianjin University)
Tail estimates of heat kernels on doubling spaces.

II. Tail estimate of heat semigroup and off-diagonal upper estimate of heat kernel

 

Tuesday 14.12.21  9:15-10:45  
 
Eryan Hu  (Tianjin University)
Tail estimates of heat kernels on doubling spaces.

III. Hölder continuity and off-diagonal lower estimate of heat kernel

 

Tuesday 21.12.21  10:15-11:45  
 
Simon Nowak  (Bielefeld)
Improved Sobolev regularity for nonlocal equations with VMO coefficients

 

Tuesday 11.01.22  10:15-11:45  
 
Xinxing Tang (YMSC, Tsinghua University)
A generalized join of digraphs and path homology

 

Tuesday 18.01.22  10:15-11:45  
 
Xinxing Tang (YMSC, Tsinghua University)
A generalized join of digraphs and path homology II

 

Tuesday 25.01.22  10:15-11:45   


Philpp Sürig  (Bielefeld)
Heat kernel estimates on manifolds with ends

 

Tuesday 01.02.22  15:15-16:45  
 
Radek Wojciechowski  (CUNY)
Graphs and discrete Dirichlet spaces

 

Sommersemester 2021

The seminar meets in this semester online via Zoom.  

Tuesday 20.04.21  10:15-11:45  
 
Meng Yang  (Lisbon)
Gradient Estimate for the Heat Kernel on Some Fractal-Like Cable Systems

 

 

Tuesday 27.04.21  10:15-11:45  
 
Mael Landsade  (Nantes)
Lower bound of the spectrum of Schrödinger operators on Riemannian manifolds

 

 
 

Tuesday 04.05.21  10:15-11:45  
 
Yuhua Sun  (Nankai)
An almost sharp Liouville principle for Dmu+up|4u|q60 on geodesically complete noncompact Riemannian manifolds

Abstract. We establish an almost sharp Liouville principle for the weak solutions to the aforementioned differential inequality on geodesically complete noncompact Riemannian manifolds for the following range of parameters:  m > 1 while p and q are arbitrary real. The results is entirely new for negative p and q, even in the Euclidean spaces.  

 

Tuesday 18.05.21  10:15-11:45  
 
Alexander Bendikov  (Wroclaw)
Hierarchical Schrödinger-type operators: the case of potentials with local singularities I

 

 
 

Tuesday 25.05.21  10:15-11:45  
 
Rostislav Grigorchuk  (Texas A&M University)
Spectra on group and graphs: a short survey

Abstract. In my talk I will touch on such topics as shape of the spectrum of Cayley and Schreier graphs of finitely generated groups, type of spectral measures, the question of A.Valette "Can one hear the shape of a group", and the relation to the random Schrödinger operator. 
Based on numerous results with coauthors: L. Bartholdi, A.Zuk, Z.Sunic, D.Lenz, T.Nagnibeda, A.Perez, B.Simanek, A.Dudko and others.

 

 

Tuesday 01.06.21  10:15-11:45  

Alexander Bendikov  (Wroclaw)
Hierarchical Schrödinger-type operators: the case of potentials with local singularities II

 

 
 

Tuesday 15.06.21  13:15-14:45  
 
Leandro Pessoa   (Universidade Federal do Piauí, Brazil)

Stochastic half-space theorems for minimal surfaces and H-surfaces of R³

 

Abstract: We will talk about a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of R³. We also consider the n-dimensional case and show that any minimal hypersurface immersed with bounded curvature in M×R+ equals some slice M×{s} provided M is a complete, recurrent n-dimensional Riemannian manifold with non-negative Ricci curvature and whose sectional curvatures are bounded from above. For H-surfaces we prove that a stochastically complete surface M cannot be in the mean convex side of a H-surface N embedded in R³ with bounded curvature if

sup |HM | < H, or dist(M, N) = 0 when sup |HM | = H. Finally, we will show a maximum principle at infinity for the case where M has non-empty boundary.

This is a joint work with G.P. Bessa and L.P. Jorge (Federal University of Ceará - Brazil).

 

Wintersemester 2020/21

The seminar meets in this semester online via Zoom.  

Tuesday 01.12.20  10:15-11:45  
 
Eryan Hu  (Tianjin)
Tail estimates of heat kernels on doubling spaces

Abstract. We derive the parabolic L2-mean value inequality from the Faber-Krahn inequality, the generalized capacity condition, and the integrated jump kernel upper bound, for any regular Dirichlet form without a killing part on the doubling space. As an  application, we obtain first the on-diagonal upper bound of the heat kernel, and then the lower bound of the mean exit time on any ball, and finally the off-diagonal upper bound of the heat kernel. Our result covers two extreme cases: one is the weak upper estimate of the heat kernel corresponding to the (weakest) jump kernel upper bound in L1-norm (for example, on the ultra-metric space), and the other is the stable-like estimate of the heat kernel corresponding to the (strongest or pointwise) jump kernel upper bound in L1-norm (on the general metric space), and therefore unifies the existent achievements in this direction. We also obtain the localized lower bound of heat kernel under the Poincare inequality, the generalized capacity condition, and the (weakest) integrated jump kernel upper bound. If the Poincare inequality is replaced by the full lower bound of the jump kernel, then we can obtain the full lower bound of heat kernel, which matches the aforementioned stable-like upper bound of the heat kernel.

 
Tuesday 08.12.20  10:15-11:45  
 
Alexander  Tyulenev  (Steklov Institute, Moscow)
Restriction and extension theorems for the Sobolev W1p(Rn)-spaces. The case 1 < p ≤ n.
 

Abstract. Let S ½ Rn be a closed nonempty set such that, for some d 2 [0; n] and " > 0, the d-Hausdorff content 
Hd1(S \ Q(x; r))¸"rd for all cubes Q(x; r) centered in x 2 S with side length 2r2(0;2].  Such sets are said to be d-thick. Given a d-thick set S ½ Rn, for each p>maxf1; n-dg we present an intrinsic characterization of the trace space W1p(Rn)jS of the Sobolev space W1p(Rn) to the set S. Furthermore, we present several interesting examples of d-thick sets and show that in some cases one can essentially simplify the corresponding criterion.

We also discus new tools and methods which are keystones for that extension problem. More precisely, we give new modifications of the classical Whitney Extension Operator, consider new  Calderon-type maximal functions and special sequences of Frostman-type measures. Our results extend those available in the case p 2 (1; n] for Ahlfors-regular sets S.

 
 
Tuesday 15.12.20  10:15-11:45  
     
Simon Nowak  (Bielefeld)
Regularity theory for nonlocal equations with VMO coefficients I
 
 
 

Tuesday 22.12.20  10:15-11:45  
    

Simon Nowak  (Bielefeld)
Regularity theory for nonlocal equations with VMO coefficients II
 
 

Tuesday 19.01.21  15:15-16:45  
 
Sergei Foss  (Novosibirsk State University and Heriot-Watt University)
Baras-Erdos graphs with random weights and perfect simulation

 

Abstract. We discuss regenerative properties of a directed acyclic random graph on the line and the algorithm for simulating the growth rate of its maximal paths. We also comment on various directions of generalisations, with introducing random weights of edges, replacing the line by a partially ordered set, etc.

 
 
Tuesday 26.01.21  10:15-11:45  
 
Philipp Sürig   (Bielefeld)
Heat kernel's lower bounds and volume growth
 
  
 
Tuesday 02.02.21  10:15-11:45  
 
Shilei Kong   (Bielefeld)
Near-isometries of hyperbolic graphs and bi-Lipschitz embeddings of their boundaries
 
 
  
Tuesday 09.02.21  10:15-11:45  
 
Liguang Liu   (Renmin University)
Hardy’s inequality and Green Function on metric measure spaces
 

 

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