SFB 1283, project
A3, IRTG 2235, area D
The seminar meets online via Zoom. Time is given in Central European Time Zone.
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Tuesday 25.10.22 10:00-11:30 Overview of path homology theory of digraphs
II |
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Tuesday 01.11.22 9:00-10:30 Overview of path
homology theory of digraphs II 2. Trapezohedra
and structure of the chain space W3. |
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Tuesday 08.11.22 9:00-10:30 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. |
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Tuesday 15.11.22 9:00-10:30 Overview of path
homology theory of digraphs II |
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Tuesday 22.11.22 9:00-10:30 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 |
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Tuesday 29.11.22 9:00-10:30 Simplicial
approach to path homology of quivers, subsets of groups and submodules of
algebras II |
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Tuesday 06.12.22 9:00-10:30 Simplicial approach
to path homology of quivers, subsets of groups and submodules of algebras III |
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Tuesday 13.12.22 9:00-10:30 Magnitude homology
and path homology Abstract: We give account of the article Yasuhiko
Asao of the same name. |
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Tuesday 20.12.22 9:00-10:30 First Betti number of the path homology
(and directed flag complex) of random directed graphs
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Tuesday 10.01.23 9:00-10:30 Connection between magnitude homology and
path homology of digraphs
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Tuesday 17.01.23 9:00-10:30 Magnitude homology
of digraphs
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Tuesday 31.01.23 9:00-10:30 Overview of path
homology theory of digraphs II 3. Combinatorial
curvature of n-cube |
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Tuesday 07.02.23 9:00-10:30 Overview of path
homology theory of digraphs II |
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Tuesday 14.02.23 9:00-10:30 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. |
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Tuesday 21.02.23 9:00-10:30 Cofibration
category of digraphs for path homology |
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Tuesday 28.02.23 9:00-10:30 Discrete Ricci curvature
and homology |
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Tuesday 14.03.23 9:00-10:30 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. |
The seminar meets online via Zoom. Time is given in Central European Time
Zone.
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Friday 25.02.22 7:00-8:30 Overview of path
homology theory of digraphs (1) |
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Friday 04.03.22 7:00-8:30 Overview of path
homology theory of digraphs (2) |
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Friday 11.03.22 7:00-8:30 Overview of path
homology theory of digraphs (3) |
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Friday 25.03.22 7:00-8:30 Overview of path
homology theory of digraphs (4) |
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Tuesday 05.04.22 9:00-10:30 Overview of path
homology theory of digraphs (5) |
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Tuesday 12.04.22 9:00-10:30 Overview of path
homology theory of digraphs (6) |
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Tuesday 19.04.22 9:00-10:30 Overview of path
homology theory of digraphs (7) |
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Tuesday 26.04.22 9:00-10:30 Overview of path
homology theory of digraphs (8) |
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Tuesday 10.05.22 9:00-10:30 Overview of path
homology theory of digraphs (9) |
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Tuesday 17.05.22 9:00-10:30 Overview of path
homology theory of digraphs (10) |
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Tuesday 24.05.22 9:00-10:30 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). |
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Tuesday 31.05.22 10:00-11:30 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. |
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Tuesday 07.06.22
10:00-11:30 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. |
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Tuesday 14.06.22 10:00-11:30 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. |
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Tuesday 21.06.22 9:00-10:30 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. |
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Tuesday 28.06.22 10:00-11:30 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. |
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Tuesday 05.07.22 10:00-11:30 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. |
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Tuesday 12.07.22 10:00-11:30 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. |
The seminar meets in this semester online via Zoom. Time is given in Central
European Time Zone.
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Tuesday 16.11.21 10:15-11:45 |
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Tuesday 23.11.21 10:15-11:45 |
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Tuesday 30.11.21
9:15-10:45 I. Parabolic mean value inequality and on-diagonal upper estimate of heat kernel |
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Tuesday 07.12.21
9:15-10:45 II. Tail estimate of heat semigroup and off-diagonal upper estimate of heat kernel |
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Tuesday 14.12.21
9:15-10:45 III. Hölder continuity and off-diagonal lower estimate of heat kernel |
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Tuesday 21.12.21
10:15-11:45 |
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Tuesday 11.01.22
10:15-11:45 |
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Tuesday 18.01.22
10:15-11:45 |
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Tuesday 25.01.22
10:15-11:45
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Tuesday 01.02.22
15:15-16:45 |
The seminar meets in this semester online via Zoom.
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Tuesday 20.04.21
10:15-11:45
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Tuesday 27.04.21
10:15-11:45
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Tuesday 04.05.21
10:15-11:45 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. |
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Tuesday 18.05.21
10:15-11:45
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Tuesday 25.05.21
10:15-11:45 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.
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Tuesday 01.06.21 10:15-11:45 Alexander Bendikov (Wroclaw)
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Tuesday 15.06.21
13:15-14:45 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). |
The seminar meets in this semester online via Zoom.
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Tuesday 01.12.20
10:15-11:45 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. |
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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 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. |
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Tuesday 15.12.20
10:15-11:45 Simon Nowak (Bielefeld) Regularity theory for nonlocal equations with VMO coefficients I |
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Tuesday 22.12.20
10:15-11:45 Regularity theory for nonlocal equations with VMO coefficients II |
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Tuesday 19.01.21
15:15-16:45
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. |
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Tuesday 26.01.21
10:15-11:45 Philipp Sürig (Bielefeld) Heat kernel's lower bounds and volume growth |
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Tuesday 02.02.21
10:15-11:45 Shilei Kong (Bielefeld) Near-isometries of hyperbolic graphs and bi-Lipschitz embeddings of their boundaries |
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Tuesday 09.02.21
10:15-11:45 Liguang Liu (Renmin University) Hardy’s inequality and Green Function on metric measure spaces |