Vietoris-Rips Complexes and Applied Algebraic Topology Seminar

Applied Algebraic Topology Network via YouTube

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Explore advanced topics in applied algebraic topology through this comprehensive seminar series focusing on Vietoris-Rips complexes and their connections to quantitative topology. Delve into cutting-edge research presentations covering Čech complexes, nerve lemmas, manifold reconstruction, Kuratowski embeddings, metric thickenings, and the tight span. Examine theoretical foundations including the filling radius, Gromov-Hausdorff distances, optimal transport, and homotopies of bounded size while discovering applications in geometric group theory, topological combinatorics, and geometric topology. Learn from leading researchers as they present their latest findings on persistent homology, injective metric spaces, geometric approaches to data analysis, and computational aspects of topological data analysis. Investigate specialized topics such as Hausmann's conjecture, metric gluings, projective codes, clique complexes, and simplicial complex decompositions. Study advanced concepts including Urysohn width, isoperimetric inequalities, scalar curvature, density-sensitive filtrations, and persistent homotopy groups. Gain insights into practical applications through discussions of Lipschitz extensions, parameterized filtrations, Gromov hyperbolicity, configuration spaces, and barcode computation complexity. Explore connections to differential geometry through topics on curvature bounds, geodesics, shape theory, and topological simplification of surfaces, while examining discrete structures, hypercube graphs, information theory applications, and matrix-valued optimization techniques.

Syllabus

Henry Adams (3/12/21): Vietoris-Rips thickenings: Problems for birds and frogs
Facundo Mémoli: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
Baris Coskunuzer (4/14/21): Geometric Approaches on Persistent Homology
Žiga Virk (4/24/21): A counter-example to Hausmann's conjecture
Bei Wang (5/7/21): Homotopy Types of Vietoris–Rips Complexes of Metric Gluings
Matthew Zaremsky (5/21/21): Vietoris-Rips complexes and geometric group theory
Florian Frick (6/4/21): Rips complexes, projective codes, and zeros of odd maps
Mikhail Katz (7/14/20): Collapsing surfaces
Jonathan Barmak: Star clusters in clique complexes and the Vietoris-Rips complex of planar sets
Francesca Tombari (8/27/21): Decomposing simplicial complexes (without losing pieces)
Yevgeny Liokumovich (9/10/21): Urysohn width, isoperimetric inequalities and scalar curvature
Sunhyuk Lim (9/24/21): The Gromov-Hausdorff distance between spheres
Jürgen Jost (10/29/21): Geometry and Topology of Data
Johnathan Bush (11/5/21): Maps of Čech and Vietoris–Rips complexes into euclidean spaces
Luis Scoccola (12/5/21): Density-sensitive and robust Vietoris-Rips filtrations
Ling Zhou (1/21/22): Persistent homotopy groups of metric spaces
Don Sheehy (2/4/21): Persistent Homology of Lipschitz Extensions
Bradley Nelson (2/19/22): Parameterized Vietoris-Rips Filtrations via Covers
Ulrich Bauer (3/4/22): Gromov hyperbolicity, geodesic defect, and apparent pairs in Rips filtrations
Fedor Manin (3/19/22): Linear nullhomotopies of maps to spheres
Stéphane Sabourau (4/1/22): Macroscopic scalar curvature and local collapsing
Henry Adams (3/22/22): Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Hannah Alpert (4/15/22): Homology of sliding-squares configuration spaces
Barbara Giunti (4/29/21): Average complexity of barcode computation for Vietoris-Rips filtrations
Regina Rotman (5/28/22): Curvature bounds and the length of the shortest closed geodesic
Diego Mondéjar Ruiz (6/10/22): Approximation of compact metric spaces by finite samples
Krystyna Kuperberg (7/22/22): Shape Theory: Vietoris-Cech approach to homotopy
Alexander Rolle (8/12/22): Homology inference for the degree-Rips bifiltration
Francisco Martinez Figueroa (8/19/22): Chromatic number of G-Borsuk graphs
Dmitri Burago (9/16/22): A Mozaic from Geometry, Dynamics, PDEs, and maybe more
Lori Ziegelmeier: Minimal Cycle Representatives in Persistent Homology using Linear Programming
Abigail Hickok 11/11/22: Persistence Diagram Bundles: A multidimensional generalization of vineyards
Éric Colin de Verdière (12/9/22): Optimal topological simplification of surfaces
Sergio Zamora (1/20/23): The lower semi-continuity of \pi_1 and nilpotent structures in persistence
Urs Lang (2/3/23): Combinatorial dimension and higher-rank hyperbolicity
Irina Gelbukh 2023: The Reeb graph of a smooth function encodes the function class and manifold type
Emilie Purvine (3/3/23): Applied Topology for Discrete Structures
Nicolò Zava (3/17/23): Every stable invariant of finite metric spaces produces false positives
Samir Shukla (3/31/23): Vietoris-Rips complexes of hypercube graphs
Hubert Wagner (4/14/23): Topology of... surprisal: Information theory and Vietoris-Rips filtrations
Uzu Lim (5/26/23): Strange random topology of the circle
Alexey Balitskiy (6/9/23): Urysohn width
Melanie Weber (6/16/23): Exploiting geometric structure in matrix-valued optimization
Shin-ichi Ohta (6/23/23): Barycenters and a law of large numbers in Gromov hyperbolic spaces
Eugene Stepanov (7/7/23): Reconstructing hidden geometric structures in data from distance matrices