# Oberseminar Geometric Analysis

SFB 1283, project A3,  IRTG 2235, area D

# Wintersemester 2022/23

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

 Tuesday 25.10.22  10:00-11:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs II 1. Structure of the chain space W2.

 Tuesday 01.11.22  9:00-10:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs II 2. Trapezohedra and structure of the chain space W3.

 Tuesday 08.11.22  9:00-10:30     Xiang Liu (Nankai University & BIMSA, Beijing) Computing persistent path homology of digraphs Abstract: Path homology is a great tool to characterize the assymetric structure of data. However, the lack of efficient algorithms hinders the applications of this great tool. In this talk, we will introduce our recently developed algorithms for persistent path homology of digraphs. More specifically, we will give algorithms for general persistent embedded homology of graded groups in chain complexes over field coefficient from chain level. By applying the general algorithms to digraphs and hypergraphs, efficient algorithms for persistent path homology of digraphs and persistent embedded homology of hypergraphs can be derived. The derived algorithms have smaller time complexity than existing algorithms. We will also give performance comparison between our algorithm and existing ones. It shows that our algorithm is very efficient.

 Tuesday 15.11.22  9:00-10:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs II 2. Trapezohedra and structure of the chain space W3 (continued)

 Tuesday 22.11.22  9:00-10:30     Sergei O. Ivanov  (BIMSA, Beijing) Simplicial approach to path homology of quivers, subsets of groups and submodules of algebras I   Abstract: We develop the path homology theory in a general simplicial setting which includes as particular cases the original path homology theory for path complexes and new homology theories:  homology of subsets of groups and Hochschild homology of submodules of algebras. Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor’yan, Muranov, Vershinin and Yau

 Tuesday 29.11.22  9:00-10:30     Sergei O. Ivanov  (BIMSA, Beijing) Simplicial approach to path homology of quivers, subsets of groups and submodules of algebras II

 Tuesday 06.12.22  9:00-10:30     Sergei O. Ivanov  (BIMSA, Beijing) Simplicial approach to path homology of quivers, subsets of groups and submodules of algebras III

 Tuesday 13.12.22  9:00-10:30     Shiquan Ren (Henan University) Magnitude homology and path homology   Abstract: We give account of the article Yasuhiko Asao of the same name.

Tuesday 20.12.22  9:00-10:30

Thomas Chaplin (Oxford)

## First Betti number of the path homology (and directed flag complex) of random directed graphs

Tuesday 10.01.23  9:00-10:30

Dana Melnikova (HSE, Moscow)

## Connection between magnitude homology and path homology of digraphs

 Tuesday 17.01.23  9:00-10:30     Shiquan Ren (Henan University) Magnitude homology of digraphs

 Tuesday 31.01.23  9:00-10:30     Alexander Grigor’yan  (Bielefeld and CUHK) Overview of path homology theory of digraphs II 3. Combinatorial curvature of n-cube

 Tuesday 07.02.23  9:00-10:30     Alexander Grigor’yan  (Bielefeld and CUHK) Overview of path homology theory of digraphs II 3. Combinatorial curvature of n-cube (contd.)

 Tuesday 14.02.23  9:00-10:30     Bobo Hua  (Fudan University) A graph with nonnegative Ollivier curvature has at most two ends   Abstract: Ollivier introduced a curvature notion on graphs. This was modified by Lin-Lu-Yau, see also Muench-Wojciechowski. As a corollary of Cheeger-Gromoll splitting theorem, a manifold with nonnegative Ricci curvature has at most two ends. We prove a discrete analog of the above result. In general, the Busemann function on a graph with nonnegative curvature may not be harmonic. The key ingredient of the proof is the existence of a nontrivial linear growth harmonic function if the graph has more than one end. This is based on joint work with Florentin Muench.

 Tuesday 21.02.23  9:00-10:30     Brandon Doherty  (Stockholm University) Cofibration category of digraphs for path homology

 Tuesday 28.02.23  9:00-10:30     Florentin Münch  (Leipzig) Discrete Ricci curvature and homology

 Tuesday 14.03.23  9:00-10:30     Xueping Huang  (Nanjing University of Information Science and Technology) Integral maximum principle and uniqueness class problem for graphs   Abstract: Some problems concerning the large scale analysis and geometry are closely related to the uniqueness of solutions to the heat equation. In this talk we show that a discrete analogue of the integral maximum principle method works well for the uniqueness class problem, the stochastic completeness problem and the semigroup gradient estimates under discrete Ricci lower bound.

# Sommersemester 2022

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

 Friday 25.02.22  7:00-8:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs (1)

 Friday 04.03.22  7:00-8:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs (2)

 Friday 11.03.22  7:00-8:30     Alexander Grigor’yan  (Bielefeld) Overview of path homology theory of digraphs (3)

 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).

 Tuesday 31.05.22  10:00-11:30     Jian Liu (BIMSA and Hebei Normal University) Applications of path homology in networks   Abstract:  Path homology is a powerful method for attaching algebraic invariants to digraphs.  This topic is based on the recent work ([1]) by S. Chowdhury et al. The work is to present an algorithm for path homology and use it to topologically analyse a variety of real-world complex temporal networks. A crucial step in the analysis is the complete characterization of path homologies of certain families of small digraphs that appear as subgraphs in these complex networks.   Using information from this analysis, the authors identify small digraphs contributing to path homology in dimension two for three temporal networks in an aggregated representation and relate these digraphs to network behavior. Another step is to investigate alternative temporal network representations and identify complementary subgraphs as well as behavior that is preserved across representations.   The work shows that path homology provides insight into temporal network structure, and in turn, emergent structures in temporal networks provide us with new subgraphs having interesting path homology.   In this talk, we will give a presentation for reviewing the following articles ([2,3]) together with a short report on our progress on the topic.   [1] Chowdhury S, Huntsman S, Yutin M. Path homologies of motifs and temporal network representations. Applied Network Science, 2022, 7(1): 1-23. [2] Chowdhury S, Mémoli F. Persistent path homology of directed networks. Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2018: 1152-1169. [3] Chowdhury S, Gebhart T, Huntsman S, et al. Path homologies of deep feedforward networks. 18th IEEE International Conference on Machine Learning and Applications (ICMLA). IEEE, 2019: 1077-1082.

 Tuesday 07.06.22  10:00-11:30     Mengmeng Zhang (Hebei Normal University and BIMSA) The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets   Abstract:  Different from classical homology theory,  Alexander Grigor'yan, Yuri Muranov and Shing-Tung Yau recently introduced $delta$-(co)homology,  taking the  (co)boundary homomorphisms as $\delta$-weighted alternative sum of (co)faces. For understanding the ideas of $delta$-homology, Li, Vershinin and Wu introduced $delta$-twisted homology and homotopy in 2017. On the other hand, the twisted Cartesian product of simplicial sets was introduced by Barratt, Gugenheim and Moore in 1959, playing a key role for establishing the simplicial theory of fibre bundles and fibrations. The corresponding chain version is twisted tensor product introduced by Brown in 1959.   In this talk, I will report our recent progress for unifying $delta$-homology and twisted Cartesian product. We introduce $\Delta$-twisted Carlsson construction of $\Delta$-groups and simplicial groups, whose abelianization gives a twisted chain complex generalizeing the $delta$-homology, called $\Delta$-twisted homology. We  show that Mayer-Vietoris sequence theorem holds for $\Delta$-twisted homology. Moreover, we introduce the concept of $\Delta$-twisted Cartesian product as a generalization of the twisted Cartesian product, and explore the fiber bundle structure. The notion of $\Delta$-twisted smash product, which is a canonical quotient of $\Delta$-twisted Cartesian product, is used for determining the homotopy type of $\Delta$-twisted Carlsson construction of simplicial groups.

 Tuesday 14.06.22  10:00-11:30     Wang Chong (Cangzhou Normal University) Discrete Morse Theory on Digraphs   Abstract: Discrete Morse theory is a discrete version of the classical Morse theory of smooth Morse functions on manifolds. In 1998, R. Forman invented the discrete Morse theory for simplicial complexes or general cell complexes. In the subsequent study, R. Ayala et al. studied the discrete Morse theory on graphs by using cliques as flag complexes on graphs which are analogues of simplicial complexes. Discrete Morse theory can greatly reduce the number of cells and simplices, simplify the calculation of homology groups, and can be applied to topological data analysis.       Inspired by this, based on the path homology theory of digraphs which has been initiated and studied by A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau, we discuss the discrete Morse theory on digraphs. We define the discrete Morse functions on digraphs, give the Morse complex of digraphs by quasi-isomorphism and prove that the path homology groups of digraphs and Morse complex are isomorphic. Furthermore, we give the discrete Morse inequalities on digraphs. The first part is based on the joint work with Professor Yong Lin and Professor S. T. Yau and the second part is based on the joint work with Professor Yong Lin.

 Tuesday 21.06.22  9:00-10:30     Dong Chen (Peking University, Shenzhen Graduate School, and BIMSA) Persistent path homology in molecular and material sciences   Abstract: Path homology introduced by Yau and coworkers is mathematically rich and opens new directions in both pure and applied mathematics.  As a generalization,  persistent path homology (PPH) enables a multiscale analysis of directed graphs (digraphs) and networks.  In this work,  we introduce PPH to analyze and characterize directed structures in molecular and material sciences. PPH unveils the Jahn-Teller effect and distinguishes different catalysts with the same conformation in materials science. We also propose angle-based persistent path homology to discriminate spatial isomers in molecular science, including Cis-Trans structures and chiral molecules. Additionally, angle-based PPH uncovers unique structural units with mirror symmetry that may be present in high-entropy alloys. Finally, PPH is applied  to systems biology to describe the blood coagulation formation, revealing its pivoting stages.

 Tuesday 28.06.22  10:00-11:30     Xinxing Tang (BIMSA, Tsinghua, Beijing) Minimal path and acyclic model   Abstract: I will talk about the structure of the path complex (Ω_(G;Z), ∂) via the Z-generators of Ω_*(G;Z), which is called the minimal path in Huang-Yau's paper. I will define the corresponding supporting digraph of a minimal path and prove that such supporting digraph has acyclic path homologies. Several examples of minimal path of length 3 and its supporting digraph will be given. Finally, we will talk about the basic applications of the acyclic models.

Tuesday 05.07.22  10:00-11:30

Jian Liu (BIMSA and Hebei Normal University)

## Applications of path homology in networks II

Abstract: In this talk, as a joint work with Professor Alexander Grigor’yan, I will give a theorem concerning the conjecture in the paper [1] by S. Chowdhury et al. A direct application of this result shows that any finite sub-digraph of an n-dimensional cubic network has null path homology in dimension \geq n.

I will introduce the persistent path homology, which shows application potential in biomolecule and materials science in Dong Chen’s talk. Then different homologies defined on digraphs introduced in [2,3] will be compared. The directed flag complex homology (DFC homology) and the path homology of multilayer perceptrons (MLPs) will be shown. The Dowker complex homology and path homology on a symmetric network with real-valued weights have the same persistent diagram in dimension 1.

[1] Chowdhury S, Huntsman S, Yutin M. Path homologies of motifs and temporal network representations[J]. Applied Network Science, 2022, 7(1): 1-23.

[2] Chowdhury S, Mémoli F. Persistent path homology of directed networks[C]//Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2018: 1152-1169.

[3] Chowdhury S, Gebhart T, Huntsman S, et al. Path homologies of deep feedforward networks[C]//2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA). IEEE, 2019: 1077-1082.

Tuesday 12.07.22  10:00-11:30

Anna Kh. Balci (Bielefeld)

## Variational problems and Lavrentiev gap in partial Sobolev spaces of differential forms

Abstract: We study variational problems in  generalised Sobolev-Orlicz spaces of differential forms.  In particular we provide results on density of smooth functions and design examples on Lavrentiev gap for partial spaces of differential forms. The construction is based on a Cantor type “singular set”. As the application we  demonstrate the Lavrentiev for several models including borderline case of double-phase potential. The talk based on join work with Mikhail Surnachev.

# 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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