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Research Seminars: Geometry & Topology

Spring 2021

Time & Location: All talks are on Thursday in Gibson Hall 308 at 12:30 PM unless otherwise noted.
Organizer: Komendarczyk, Rafal



March 12

Title: Vietoris-Rips thickenings: Problems for birds and frogs

Henry Adams | Colorado State

Abstract: An artificial distinction is to describe some mathematicians as birds, who from their high vantage point connect disparate areas of mathematics through broad theories, and other mathematicians as frogs, who dig deep into particular problems to solve them one at a time. Neither type of mathematician thrives without the help of the other! In this talk, I will survey open problems related to Vietoris-Rips complexes that are attractive to both birds and frogs. Though Vietoris-Rips complexes are frequently used to approximate the shape of a dataset, many questions remain about their mathematical properties. Frogs may delight in open problems such as the homotopy types of Vietoris-Rips complexes of spheres, ellipsoids, tori, graphs, Cayley graphs of groups, geodesic spaces, subsets of the plane, and even the integer lattice Z^n with the taxicab metric for n >= 4. Birds may enjoy emerging connections between Vietoris-Rips complexes and a variety of areas in pure mathematics, including metric geometry (Gromov-Hausdorff distances), quantitative topology (Gromov's filling radius), measure theory (optimal transport), topological combinatorics (Borsuk-Ulam theorems), geometric group theory (finiteness properties of groups), and geometric topology (thick-thin decompositions).

Time: 10:00am CT


March 26

Title: AATRN: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius

Facundo Mémoli | Ohio State

Abstract: The persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. We consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space X into a larger ambient metric space E and then considering neighborhoods of the original space X inside E.

We then prove that the persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space E satisfies a property called injectivity.

As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces.

Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants.

As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M. Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F. Wilhelm.

Time: 10:00am CT


March 31

Title: AATRN: Graph representation learning and its applications to biomedicine

Marinka Zitnik - Harvard University

Abstract: The success of machine learning  depends heavily on the choice of representations used for prediction tasks. Graph representation learning has emerged as a predominant choice for learning representations of networks. In this talk, I describe our efforts to expand the scope and ease the applicability  of graph representation learning. First, I outline SubGNN, a subgraph neural network for learning disentangled subgraph embeddings. SubGNN generates embeddings that capture complex subgraph topology, including structure, neighborhood, and position of subgraphs  in a graph. Second, I will discuss applications in biology and medicine. The new methods predicted disease treatments that were experimentally verified in the wet laboratory. Further, the methods helped to discover dozens of combinations of drugs safe for  patients with considerably fewer unwanted side effects than today's treatments. Lastly, I describe our efforts in learning actionable representations that allow users of our models to receive predictions that can be interpreted meaningfully.

Recordings of most of the talks will be posted to the AATRN YouTube Channel

Time: 10:00am CT


April 7

Title: AATRN: How optimal transport can help us to determine the curvature of complex networks?

Marzieh Eidi | Max Planck

Abstract: Ollivier Ricci curvature  is a notion that originated from Riemannian Geometry and is suitable for applying on different settings from smooth manifolds to discrete structures such as (directed) hypergraphs. In the past few years, alongside Forman Ricci curvature, this curvature as  an edge-based measure has become a popular and powerful tool for network analysis. This notion is defined based on optimal transport problem (Wasserstein distance) between sets of probability measures supported on data points and can nicely detect some important  features such as clustering and sparsity in their structures. After introducing this notion for (directed) hypergraphs and mentioning some of its properties, as one of the main recent applications, I will present the result of the implementation of this tool  for the analysis of chemical reaction networks. 

Recordings of most of the talks will be posted to the AATRN YouTube Channel.

Time: 10:00am CT


April 13

Title: AATRN: High-dimensional data, level-set  geometry, and Voronoi analysis of spatial point sets

Menachem Lazar | Bar-Ilan University

Abstract: Physical systems are regularly  studied as spatial point sets, and so understanding the structure of such sets is a very natural problem.  However, aside from special cases, describing the manner in which a set of points is arranged in space can be quite challenging.  In the first part of  this talk, I will show how consideration of the configuration space of local arrangements of neighbors can shed light on essential challenges of this problem, and in the classification of high-dimensional data more generally.  In the second part of the talk  I will introduce some ideas from Voronoi cell topology and show how they can be used to define crystals, defects, and order more generally in a somewhat precise manner. 

Recordings of most of the talks will be posted to the AATRN YouTube Channel.

Time: 10:00am CT