# Autumn 2020

** Sep 10th. Sunhyuk Lim, ** Convex regions on the n-dimensional spherical surface; ** Facundo Memoli, ** Remarks and ideas about Vietoris-Rips complexes.

** Sep 17th. Qingsong Wang, ** Rips complexes as nerves and a Functorial Dowker-Nerve Diagram.

** Sep 24th. Mario Gomez Flores, ** Diameter-Extremal Subsets of Spheres.

** Oct 1st. Marco Contessoto, ** Persistent Cohomology and Circular Coordinates.

** Oct 8th. Musashi Koyama, ** Algorithm for persistent homology.

** Oct 15th. Ling Zhou, ** Ripser: efficient computation of Vietorisâ€“Rips persistence barcodes.

** Oct 22nd. Kun Jin, ** NLP overview.

** Oct 29th. Marco Contessoto, ** Generalized persistence diagrams.

** Nov 5th. Nate Clause, ** TBD.

** Nov 12nd. Kun Jin, ** NLP overview; ** Anastasios Stefanou, ** Universality of persistence diagrams and the bottleneck and Wasserstein distances.

** Nov 19th. Zhengchao Wan, ** A flow tangent to the Ricci flow via heat kernels and mass transport.

** Dec 3rd. Luis Scoccola, ** TBD.

# Summer 2020

** May 28th. Musashi Koyama, ** Persistence by Parts: Multiscale Feature Detection via Distributed Persistent Homology.

** Jun 4th. Nate Nathaniel, ** On rectangle-decomposable 2-parameter persistence modules.

** Jun 11th. Musashi Koyama, ** Persistence by Parts: Multiscale Feature Detection via Distributed Persistent Homology.

** Jun 18th. Woojin Kim, ** Elder-Rule-Staircodes for Augmented Metric
Spaces

** Jun 25th. Marco Contessoto, ** Persistent cohomology and cup length invariant.

** Jul 2nd. Anastasios Stefanou and Woojin Kim, ** Modules over posets: commutative and homological algebra.

** Jul 9th. Musashi Koyama (candidacy practice), ** Methods of calculating homology in a distributed fashion; ** Anastasios Stefanou, ** Research lecture overview.

** Jul 16th. Anastasios Stefanou, ** Research Lecture.

** Jul 23rd. Sunhyuk Lim, ** Topological representation of the geometry of metric spaces.

** Aug 6th. Zhengchao Wan and Kun Jin, ** Ricci flow.

** Aug 13th. Mario Gomez Flores and Qingsong Wang, ** Finite subset spaces of S^1.

** Aug 20th. Marco Contessoto,** Persistent diagrams as diagrams.

** Aug 27th. Ling Zhou, ** Multiplicative persistent distances.

# Spring 2020 at 1:45 PM in JH 382

** Jan 9th. Zhengchao Wan, ** Ultrametics and monoidal metric spaces.

** Jan 16th. Marco Contessoto (20 mins), ** State of the art on persistent co-homology; ** Sunhyuk Lim, ** Brief review of classical MDS.

** Jan 23rd. Qingsong Wang, ** Persistent barcode and Laplace eigenfunction on surface.

** Jan 30th. Alessandro Gregorio (20 mins), ** A concept of weak isometry for finite metric spaces and related invariants; ** Musashi Koyama, ** Spectral Sequence.

** Feb 6. Marco Contessoto (20 mins), ** Why is persistent cohomology faster than persistent homology? ** Anastasios Stefanou, ** A_\infty persistent homology estimates the topology from point cloud datasets.

** Feb 13th. Anastasios Stefanou, ** A_\infty persistent homology estimates the topology from point cloud datasets. ** Nate Nathaniel, ** Random coagulation/fragmentation process.

** Feb 20th. Ling Zhou, ** Persistent homotopy groups of metric spaces.

** Feb 27th. Marco Contessoto (20 mins), ** TBD. ** Kun Jin, ** The Gaussian Transform.

** Mar 5th. Marco Contessoto, ** Why is persistent cohomology faster than persistent homology?

** Mar 12th. ** Spring Break.

** Mar 19th. Marco Contessoto (20 mins), ** TBD; ** Mario Gomez Flores, ** Persistence as an approach to curvature sets.

** Mar 26th. Woojin Kim, ** Metrics and Persistent Homology for Dynamic Geometry; ** Kritika Singhal, ** Methods of Simplification of Metric spaces, and Lower bounds on the Gromov-Hausdorff distance.

** Apr 2nd. Marco Contessoto (20 mins), ** TBD; ** Zhengchao Wan, ** Ricci flow.

** Apr 9th. Zhengchao Wan, ** Ricci flow (Continued).

** Apr 16th. Musashi Koyama, ** Distributing persistent homology via spectral sequences.

** Apr 23th. Marco Contessoto, ** TBD.

** Apr 30th. Ling Zhou, ** Rational homotopy and Sullivan minimal models.

** May 7th. Musashi Koyama, ** Distributing persistent homology via spectral sequences (Continued).

** May 14th. Marco Contessoto (20 mins), ** TBD; ** Zhengchao Wan, ** Gromov Hausdorff geodesic are Hausdorff realizable.

# Autumn 2019 at 1:45 PM in JH 382

** August 22. Zhengchao Wan (1 hour); ** An estimate for the sum of Betti numbers ** Ling Zhou (15-20 mins); ** Vietoris-Rips complex of finite tree metric spaces. ** Qingsong Wang (40 mins) **; Loops in Reeb graphs

** August 29. Ling Zhou (40 mins); ** Vietoris-Rips complex of finite tree metric spaces. ** Facundo (20mins); ** More remarks added to Zhengchao and Ling's talks, **
(20-30 mins) ;** Optimal Transport Survey

** September 5. Qingsong Wang (40 -50 mins) **; Loops in Reeb graphs ** Hao Xing (45 mins) **; Vector diffusion maps and the connection Laplacian ;

** September 12. Sunhyuk Lim (1.5 hours) **; On the filling radius of positively curved manifolds

** September 19. Anastasios Stefanou (30 mins) **; The Brouwer's Fixed point theorem and its applications, ** Qingsong Wang (30 mins) **; On homotopy types of Vietoris-Rips complexes of metric gluings, ** Ling Zhou (30 mins) **; Gromov-Hausdorff distances between cycle graphs and star graphs.

** September 26 (JH 355) . Musashi Koyama (15 mins) **; Homological Algebra for Persistence Modules , ** Daryl DeFord **; TBD

** October 3. ** Organization for the rest of the semester

** October 10. ** Autumn Break

** October 17. Musashi Koyama (15 mins) **; Homological Algebra for Persistence Modules , ** Kun Jin (20-30 mins) **; Updates in his project

** October 24. Musashi Koyama ( ~ 30 mins) **; Homological Algebra for Persistence Modules , ** Mario Gomez Flores ( ~30 mins) **; Nash equilibrium.

** October 31. Kritika Singhal (1 hour) **; A Mass Transform.

** November 7. In Caldwell Lab 0183 Nate Nathaniel (30 mins) **; Computational implementation of the erosion diatance, ** Kun Jin (~ 30 mins) **; Updates in his project, ** Sunhyuk Lim (~ 30 mins) ; ** Intersection of balls on the constant curvature spaces.

** November 14. Ling Zhou **; Persistent Fundamental Groups of Metric Spaces

** November 21. Mario Gomez Flores (~1 hour) **; TBD

** December 5. Musashi Koyama (1 hour) **; Homological Algebra for Persistence Modules

# Summer 2019

at 1:45 PM in CH 240** May 2. Woojin Kim **; Relationship between the rank functions and the barcodes for persistence modules over general posets.

** May 9. Woojin Kim **; Conley index and Connection matrice

** May 16. Osman Okutan **; Persistence, Metric Invariants and Simplification (Thesis defense practice)

** June 6. Hao Xing **; UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

** June 13. Ling Zhou **; Spanning Trees and fundamental groups of graphs

** June 20. Hao Xing **; PERSISTENCE BARCODES AND LAPLACE EIGENFUNCTIONS ON SURFACES

** June 27. Zhengchao Wan **; Topology and Curvature of Metric Spaces

** July 11. Nathaniel Clause **; Classification of Flocking Behavior via Zigzag Persistent Homology

** July 18. Hao Xing **; Persistent homology and Floer-Novikov theory

** July 24. Anastasios Stefanou **; Magnitude of metric spaces and magnitude homology 1 .

** July 25. Kritika Singhal **; On hyperbolicity constant

** July 26. Hao Xing **; Persistence modules with operators in Morse and Floer theory

** August 1. Anastasios Stefanou **; Magnitude of metric spaces and magnitude homology 2.

In Jennings Hall 382 below

** August 8. Kun Jin **; Comparisons of Meanshift, Gaussian Transform and Wasserstein Transform

** August 15. Anastasios Stefanou **; Magnitude of metric spaces and magnitude homology 3. ** Zhengchao Wan ; ** An estimate for the sum of Betti numbers

# Spring 2019 at 1:45 PM in CH 240

** Jan 10. ** ** Samir Chowdhury **; Graph distance from the topological view of non-backtracking cycles

** Jan 17. ** ** Sunhyuk Lim ** ; Metric thickenings of Euclidean submanifolds, Metric reconstruction via optimal transport

** Jan 24. ** ** Zhengchao Wan **; A geometric study of Wasserstein spaces: isometric rigidity in negative curvature

** Jan 31. ** ** Simon Zhang **; Dualities in persistent (co) homology

** Feb 7. ** ** Amit Patel (Colorado State Univ) **; Smoothing Operator

** Feb 14. ** ** Zhengchao Wan **; A geometric study of Wasserstein spaces, ** Kritika Singhal **; Borsuk-Ulam type theorems for metric spaces.

** Feb 21. ** ** Scott Newton **; A derived isometry theorem for constructible sheaves on R

** Feb 28. Anastasios Stefanou ** ; Computation of the interleaving distance between multidimensional persistence

** Mar 7. Woojin Kim ** ; (1) Stable Persistent Homology Features of Dynamic Metric Spaces and (2) Rank Invariant for Zigzag Modules

** Mar 14. Spring Break **

** Mar 28. (1) Hao Xing, ** ; Generalized Persistence Diagrams , ** (2) Kritika Singhal ** ; Box distance between special orthogonal groups

** April 4. Anastasios Stefanou ** ; Computation of the interleaving distance between multidimensional persistence

** April 11. Anastasios Stefanou **; Computation of the interleaving distance between multidimensional persistence

** April 18.** Plan for next semester; paper assignment

** April 25. Guilherme Vituri Fernandes Pinto **

** TBD. ** ** Sunhyuk Lim ** ; Limit theorems for persistence diagrams

# Autumn 2018 at 1:45 PM in CH 240

** Aug 30. ** ** Woojin Kim **; Multiscale analysis of dynamic metric spaces via Cosheaves.

** Sep 6. ** Planning for Autumn 2018 and Spring 2019.

** Sep 13. ** ** Ling Zhou **; Interval Decomposition of Infinite Zigzag Persistence Modules

** Sep 20. ** ** Anastasios Stefanou **; "Dynamics on Categories" to include introductory material on monoidal actions on categories, also categories with a flow and equivariant functors, and also methods from TDA that can be realized as such. "The Functoriality of the Interleaving Distance" where I will define the interleaving distance and show it is an extended pseudometric, show that flow equivariant functors are 1-Lipschitz and give examples of interleaving distances and stability results from TDA along the way.

** Sep 27. ** ** Ying Yin **; Persistent homology on teeth classification, ** Osman Okutan **; Hyperbolic Metric Spaces 1

** Oct 4. ** ** Zhengchao Wan **; Geometry of the Space of Phylogenetic Trees

** Oct 11. Autumn Break **

** Oct 18. **** Ying Yin **; Persistent homology on teeth classification, ** Anastasios Stefanou **; "The Geometry of Phylogenetic Networks" where I will discuss a recent result that the $\ell^infty$-cophenetic metric on phylogenetic trees can be realized as an interleaving distance and discuss future directions for research, e.g. define properly a category of phylogenetic networks and devising a metric for comparison for these structures which is an important open problem in computational biology.

** Oct 25. ** ** Scott Newton **; Universality of the Homotopy Interleaving Distance

** Nov 1. ** ** Ying Yin **; Persistent homology on teeth classification, ** Sunhyuk Lim **; Metric thickenings of Euclidean submanifolds, Metric reconstruction via optimal transport

** Nov 8. ** ** Woojin Kim **; Generalized persistence analysis based on stable rank invariant, Multidimensional Persistence and Noise, Stable Invariants for Multidimensional Persistence

** Nov 15. ** ** Ying Yin **; Persistent homology on teeth classification, ** Kritika Singhal **; Hyperbolic Metric spaces 2

** Nov 29. ** ** Guilherme Vituri Fernandes Pinto **; A persistent Kunneth formula for an unusual product of metric (or not) spaces.

** Dec 6. ** ** Ying Yin **; Persistent homology on teeth classification, ** Ling Zhou **; The Structure and Stability of Persistence Modules

# Spring, Summer 2018

**Jan 15, 2018.** *Osman Okutan *; Approximating metric spaces with Reeb type graphs

**Jan 22, 2018.**(1)

*Ying Yin*; Persistent homology on teeth classification, (2)

*Woojin Kim*; Algebraic stability of zigzag persistence modules. Note

**Jan 29, 2018.** (1) * Sunhyuk Lim *; Equiarea partition of general Riemannian manifolds, (2) * Woojin Kim *; Algebraic stability of zigzag persistence modules.
Note

**Feb 5, 2018.** (1)* Ying Yin *; Persistent homology on teeth classification, (2) *Osman Okutan *; Tight spans of metric spaces,

**Feb 12, 2018.** (1) *Osman Okutan *; Tight spans of metric spaces.

**Feb 19, 2018.** (1) * Ying Yin *; Persistent homology on teeth classification, (2) * Tom Needham *; Injectivity Properties of Distance Distributions,

**Feb 26, 2018.** (1) * Woojin Kim *; Algebraic Stability of Zigzag persistence modules Note ; (2) * Osman Okutan *; Tight span of metric spaces.

**Mar 5, 2018.** (1) * Woojin Kim *; Stable Siganature for Dynamic Metric Spaces via Zigzag Persistence Homology; (2) * Osman Okutan *; Reeb Posets and Tree Approximations; (3) * Ying Yin *; Persistent homology on teeth classification.

** Mar 19, 2018.** * Samir Chowdhury *; A generalization of Gromov's reconstruction theorem.

** Mar 26, 2018. ** (1) * Xiao Zha*; Topological data analysis on road network data, (2) * Ying Yin *; Persistent homology on teeth classification, (3) * Woojin Kim *; Algebraic Stability of Zigzag persistence modules.

** April 2, 2018. ** (1) *Woojin Kim *; Algebraic Stability of Zigzag persistence modules, (2) * Sunhyuk Lim *; Overview of general Markov processes.

** April 9, 2018. ** (1) * Ying Yin *; Persistent homology on teeth classification, (2) * Sunhyuk Lim *; Overview of general Markov processes, (3) * Zhengchao Wan *; Categorification of Gromov-Hausdorff Distance and Interleaving of Functors;

** April 16, 2018. ** * Samir Chowdhury *; Geodesics in the space of persistence diagrams Frechet means for distributions of persistence diagrams (In "Frechet Means for Distributions of Persistence Diagrams", one of the main results is that the space of persistence diagrams equipped with a certain L2 norm has nonnegative curvature in the sense of Alexandrov. An implicit claim in the proof of this theorem is that all geodesics in this space arise as a straight line interpolation. We supply a detailed proof of this claim, using results about the convexity of Lp norms.
)

** April 23, 2018. ** (1) * Ying Yin *; Persistent homology on teeth classification, (2) * Sunhyuk Lim *; Overview of general Markov processes.

** April 30, 2018. ** * Sunhyuk Lim *; Overview of general Markov processes.

** May 7, 2018. ** (1) * Ying Yin *; Persistent homology on teeth classification, (2) * Sunhyuk Lim *; Overview of general Markov processes, (3) * Kritica Singhal *; Sketching and Clustering Metric Measure Spaces

** May 28, 2018. ** (1) * Alex Elchesen *; Reflection distance between Zigzag modules and stability, (2) * Sunhyuk Lim *; Overview of general Markov processes.

** June 4, 2018. ** at 1 PM (1) * Kritica Singhal *; Fractal Dimension and Lower Bounds for Geometric Problems, (2) * Ying Yin *; Persistent homology on teeth classification, (3) * Sunhyuk Lim *; Overview of general Markov processes, (4) * Zhengchao Wan *; Interleaving and Gromov-Hausdorff distance.

** June 11, 2018. ** at 1:30 PM (1) * Zhengchao Wan *; Interleaving and Gromov-Hausdorff distance.

** June 18, 2018. ** at 1:30 PM (1) * Samir Chowdhury *; Persistent homology methods for asymmetric networks (30 mins), (2) * Tom Needham *; Gromov-Hausdorff distance between metric graphs.

** June 25, 2018. ** at 1:30 PM * Tom Needham *; Local Uniqueness of the Circular Integral Invariant.

** July 2, 2018. ** at 1:30 PM * Sunhyuk Lim *; Heat kernel on fractal sets.

** July 9, 2018. ** at 1:30 PM in MW 154 (1) * Simon Zhang *; GPU spectral sequence algorithm with software GPUPH

** July 16, 2018. ** at 1 PM (1) * Woojin Kim * ; Formigrams- Clustering summary of dynamic data (20 mins), (2) * Simon Zhang *; GPU spectral sequence algorithm with software GPUPH
Three References: (1) Spectral Sequences, Exact Couples and Persistent Homology of filtrations , (2) You could have invented spectral sequences , (3) Spectral sequences via examples

** July 23, 2018. ** (1) * Woojin Kim * ; Formigrams- Clustering summary of dynamic data (20 mins), (2) * Simon Zhang *; GPU spectral sequence algorithm with software GPUPH

** July 30, 2018. ** (1) * Woojin Kim * ; Formigrams- Clustering summary of dynamic data (20 mins), (2) * Osman Okutan * ; Spectral Sequences induced from the Vietories Rips filtrations.

** August 6, 2018. ** at 4 PM in CH 240 * Bowen Dai * ; The Importance of Forgetting: Limiting Memory Improves Recovery of
Topological Characteristics from Neural Data

** August 20, 2018. ** at 1:30 PM in CH 240 * Sunhyuk Lim * ; On Alexandrov curvature.