December 2, 2022

Speaker: Shu Kanazawa

Title: Large deviation principle for persistence diagrams of random cubical filtration.

Abstract

The objective of this work is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher dimensional unit cubes with integer coordinates in a Euclidean space. We first prove the strong law of large numbers for the persistence diagrams, inspired by the work of Hiraoka, Shirai, and Trinh. In this talk, we are mainly interested in the decay rate of the probability that the persistence diagram is far from the limiting measure. This is the first result on the large deviation behavior of persistence diagrams themselves. This talk is based on joint work with Yasuaki Hiraoka, Jun Miyanaga, and Kenkichi Tsunoda.

November 4, 2022

Speaker: Nik Henderson 

Title: Large-scale geometry in random environments 

Abstract

Given an infinite graph, you can weight its edges randomly to yield a random geometry. First-passage percolation deals with studying the asymptotic properties of the geometry, looking at the shapes of large balls and behavior of infinite geodesics. We'll examine some of the classical theory in Euclidean lattices, then we'll contrast some of these results with phenomena we observe in hyperbolic settings.

October 28, 2022

Speaker: Nate Clause

Title: On Approximation of 2D Persistence Modules by Interval-Decomposables .

Abstract

 Persistence modules arising from one-parameter persistent homology are nice in that they decompose as a direct sum of interval modules. This fact leads to simple computations and interpretations for one-parameter persistence modules. However, there are times when we wish to use multiple parameters, such as a scale parameter and a density parameter, to generate higher-dimensional filtrations for use in persistent homology, yielding a multi-dimensional persistence module. Unfortunately, it is a well-known fact that such modules do not always decompose as a direct sum of interval modules. In our talk, we will analyze an approach by Asashiba et al, where they use representation theory of quivers to confront this issue head-on by establishing multiple methods for approximating a 2D persistence module in an m by n grid by a pair of interval decomposable modules, and show that these approximations retain important information of the original module. 

October 21, 2022

Speaker: Scott Newton

Title: Opinion Dynamics: Beyond Vector Spaces 

Abstract

Opinion dynamics is the study of how opinions in a social network evolve over time. In this talk we will review the basics of sheaf models of opinion dynamics, and then focus on work by Hans Riess and Robert Ghrist which extends these ideas to the setting where opinions live inside lattices instead of vector spaces.

September 23, 2022

Speaker: Scott Newton 

Title: Opinion Dynamics: A gentle introduction to applied sheaf theory 

Abstract

Given a social network of individuals each holding opinions on a set of topics, how do those opinions evolve over time? This is a fantastically complicated question, and modeling such a system is an active area of research. In this talk we will explore a sequence of increasingly complex models, beginning with a vertex-weighted graph and ending with a cellular sheaf of lattices. In doing so, we will define what a cellular sheaf is and see how this notion arises naturally in this context.

September 16, 2022

Speaker: Mario Gomez 

Title: A canonical decomposition theory for metrics on a finite set 

Abstract

Given a finite metric space X, there is a canonical metric graph that contains X and provides a visualization of some geometric properties of the underlying space. This metric graph is constructed in polynomial time by writing the metric as a linear combination of "split-metrics", a set of symmetric functions that are generated by a set of partitions of the underlying set. Furthermore, this split-metric decomposition is closely related to the tight span T(X) \supset X, the smallest injective space in which X embeds. Most proofs involve a lot of casework, so I will only reproduce key arguments during the talk. Instead, I will show the key examples when |X|<=5, survey the properties of the set of partitions that generate a split-metric decomposition, and the geometric properties of the metric space that induce particularly simple split-metric decompositions. If there is time, I will also explore the connections to the tight span.

April 22, 2022

Speaker: Shreeya Behera

Title: Moving robots efficiently using the combinatorics of CAT(0) cubical complexes.

Abstract

Given a reconfigurable system X, for example, a robot moving on a grid, we can see that the possible positions of X naturally form a cubical complex S(X). Now, if we know that S(X) is a CAT(0) space, then we can find the shortest path between any two points. A purely combinatorial technique is used to prove that a state complex S(X) is a CAT(0) space. The authors illustrate this very general strategy with two examples. In this talk, we will learn about the main objects of study, i.e. the positive robotic arms, and then using the combinatorial technique we will prove that the robotic arms give rise to CAT(0) cubical complexes.

April 1, 2022

Speaker: Chris Donnay

Title: High-Throughput Screening of Nanoporous Materials with Topological Data Analysis

Abstract

In order to choose viable candidate materials, be it for carbon capture or a potential new drug, it is necessary to search immense databases. It is expensive (both financially and computationally) to perform this search. Lee et al. propose using topological properties of the materials in order to aid in this task. We will examine their proposed topological signatures, as well as discuss their use of the Mapper algorithm.

March 11, 2022

Speaker: Nate Clause

Title: Computing the Generalized Rank Invariant Via Zigzag Persistence.

Abstract

In 2018, Kim and Mémoli defined a type of generalize persistence diagram for persistence modules over an arbitrary poset. This definition is based on an extension of the classical rank invariant they define, called the generalized rank invariant. Like many invariants over higher-dimensional posets, the generalized rank invariant initially seems complex from a computational standpoint. In this talk, we will briefly review the generalized rank invariant, and then cover a new approach developed by Dey, Kim, and Mémoli which allows efficient computation of the generalized rank invariant for Z^2-indexed persistence modules via computing 1-D zigzag persistence along certain paths in the underlying poset. We will close with a short discussion of potential applications of this approach and related work.

February 18, 2022

Speaker: Aziz Guelen

Title: Diagrams of Persistence Modules Over Finite Posets

Abstract

Starting with a persistence module – a functor M : P→Vec_k for some finite poset P – we seek to assign to M an invariant capturing meaningful information about the persistence module. This is often accomplished via applying a M¨obius inversion to the rank function or birth-death function. In this talk, I will establish the relationship between the rank function and birth-death function by introducing a new invariant: the kernel function. The persistence diagram produced by the kernel function is equal to the diagram produced by the birth-death function off the diagonal and we prove a formula for converting between the persistence diagrams of the rank function and the kernel function. Moreover, the diagram assignment to the kernel functions is functorial when the morphisms between persistence modules are defined via Galois connections. This is joint work with Alex McCleary.

December 10, 2021

Speaker: Ling Zhou

Title: A new polynomial time invariant for persistent cohomology

Abstract: Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. We lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation.

December 3, 2021

Speaker: Jimin Kim

Title: Configuration spaces of repulsive particles on graphs.

Abstract: I am interested in studying the configuration spaces of n points on a graph using Morse theory. This week, I review the literature of this topic by showing a few examples of the configuration space, the discrete model of it, and the homotopy equivalence between them. Moreover, the homological dimension of the configuration space is bounded above by the number of essential vertices of the graph. We expand this result to the configuration spaces of particles on a tree when the particles have repulsive energy so that any pair of points are no closer than 2r > 0 on the tree.

November 12, 2021

Speaker: Nate Clause

Title: An Overview of Representative Cycles for Persistent Homology

Abstract: Persistent homology has proven to be a valuable analytical tool for analyzing real-world data. Data is viewed as a metric space, and then a filtration is applied to yield an increasing sequence of simplicial complexes. One applies the n-th homology functor to this sequence to get a persistence module, which decomposes nicely into a collection of intervals which can be easily visualized as a persistence barcode or a persistence diagram. Each bar in a barcode represents the lifespan of an n-dimensional homology class, or an n-dimensional void, in the data. Longer bars correspond to topological features with greater persistence, and are considered to represent more important features. While a barcode tells us the persistence of features in the dataset, it does not inherently give us any information on where each bar is coming from. In this talk, we will explore representative cycles: geometric realizations in the initial metric space of each persistence interval in the barcode. We will discuss a recent algorithm for computing representative cycles, as well as if there is such a thing as an “optimal” cycle to represent a homology class, and how one might find such a cycle.

October 8, 2021

Speaker: Francisco Martínez

Title: Topological obstructions to the chromatic number.

Abstract: In his 1978 proof of Kneser's Conjecture, Lovász initiated the study of topological obstructions to the chromatic number of a graph by introducing the neighborhood complex of a graph. This work continued with Babson and Kozlov, who introduced the Hom-complexes of a graph, and proved the neighborhood complex is homotopy equivalent to a specific Hom-complex. In this talk, we describe these cell complexes and sketch some of these proofs. We also give a similar application of Hom-complexes to bound the chromatic number of G-Borsuk graphs, when they are regarded as G-spaces.

September 24, 2021

Speaker: Mario Gomez 

Title: A reinterpretation of the 4-point condition of metric trees as a limit of a generalized Ptolemy's inequality.

Abstract: In the study of persistence diagrams of 4-point subsets of the plane, Ptolemy's inequality is the key tool that provides a bound for the possible persistence diagrams. In 1970, J.E. Valentine published generalizations of Ptolemy's inequality to spherical and hyperbolic geometries, allowing for a similar characterization of the persistence diagrams of their 4-point subsets. On the other hand, tree metric spaces have no persistent homology in non-zero dimensions (as proven by Carlsson et al.). Since metric trees can be seen as spaces with constant curvature -infty, this prompts the question of whether our bounds can recover Carlsson's result, at least in the case of 4 points. The answer is not only positive, but it also leads to a reinterpretation of the 4-point condition of metric trees as the limit of Valentine's generalization of Ptolemy's inequality.

March 5th, 2021

Speaker: Francisco Martínez

Title: Chromatic Number of Generalized Borsuk Graphs

Abstract

The Borsuk Graph is the graph with vertex set the sphere S^d, and edges {x,y} if x and y are almost antipodal. It is well known that its chromatic number is d+2, which is equivalent to Borsuk-Ulam's Theorem.  So it provides an interesting model where topological obstructions to the chromatic number are tight.  By realizing that antipodality is a Z_2 action on the sphere, we can come up with many generalizations of this model, particularly by using different group actions, or by acting on different metric spaces. In this talk I'll show you some work in progress trying to bound the chromatic number of these generalized Borsuk Graphs. The techniques used vary from topological, geometrical and combinatorial to brute force computer search.

December 4, 2020 

Speaker: Johnathan Bush (CSU)

Title: Vietoris-Rips complexes of spheres and Borsuk-Ulam theorems

Abstract

Vietoris-Rips complexes defined on manifolds are generally not well understood, despite the fact that these complexes arise naturally in the context of persistent homology. I will summarize results about Vietoris-Rips complexes, and related spaces, defined on spheres. Then, I will outline an interesting connection between these spaces and generalizations of the Borsuk-Ulam theorem.

November 20, 2020 

Speaker: Ling Zhou

Title: Filtered chain complexes and verbose barcodes

Abstract

Persistent homology is a method to extract topological information of a data set at many different geometric scales. The lifetime of topological features obtained from persistent homology can be represented using a finite collection of intervals known as persistent barcodes. However, in the calculation of persistent barcodes, ephemeral features (features that are born and dead at the same time) are not taken into account. In this talk, I will present the notion of verbose barcodes by Usher and Zhang, and show how ephemeral features provide useful information of the data in some cases.

October 30, 2020 

Speaker: Jacob Hansen

Title: Dynamics on and of Cellular Sheaves .

Abstract

Cellular sheaves are a discrete, computable instantiation of sheaf theory. They can be seen as representing systems of linear constraints parameterized by a cell complex, and possess a robust cohomology theory. This sheaf cohomology admits a representation in terms of Hodge Laplacians, which are local operators on the spaces of sheaf-valued cochains. These Laplacians are a bountiful source of new ideas for network dynamics, providing novel equilibria and types of dynamics. Both the states of individual nodes and the sheaf structure may be affected by the dynamics, which we term dynamics on and of sheaves, respectively. This talk will explore the gamut of sheaf dynamics, with a running example of opinion dynamics in a social network. Here the sheaf describes a communication structure followed by the individuals in the network, and dynamics on the sheaf represent changes in opinions, while dynamics of the sheaf represent changes in the communication structure. 

October 16, 2020 

Speaker: Jossiah Oh 

Title: Geometry of Non-Transitive Graphs .

Abstract

In this talk we examine a class of non-transitive graphs and prove some results analogous to some classic theorems about the geometry of finitely generated groups. We also show that for each finitely generated group $G$, there are continuum many pairwise non-quasi-isometric regular graphs that share several geometric properties with $G$. 

October 3, 2020 

Speaker: Zhengchao Wan 

Title: Urysohn universal ultrametric space 

Abstract

A Urysohn universal ultrametric space is an ultrametric space that satisfies both the universality and the ultra-homogeneity conditions. In this talk, I will present a novel construction of a Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric. 

February 28, 2020 

Speaker: Mario Gomez

Title: Curvature sets and persistence diagrams 

Abstract

In order to simplify the computation of the Vietoris-Rips complex of a metric space X, we can take an n point sample from the space. We can then reinterpret the Vietoris-Rips as a functor from the curvature sets of X. I will show a stability result and some computational examples. 

February 14, 2020 

Speaker: Francisco Martinez

Title: The Chromatic Number of Random Borsuk Graphs

Abstract

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in R^(d+1), and a pair of vertices is connected if the Euclidean distance between them is at least 2−ϵ. We are interested in the chromatic number of this graph as n tends to infinity. Motivated by Lovász's result from 1978 relating the topology of a graph with its chromatic number, we show that for this model topological lower bounds are tight. This contrasts with the Erdős–Rényi random graph studied by Kahle in 2007, where these bounds are not efficient. Our proof depends on combining topological methods (the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. This is joint work with Matthew Kahle.

November 1, 2019 

Speaker: Hyeran Cho

Title: Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

Abstract

A genus one 1-bridge knot (simply called a (1, 1)-knot) is a knot that can be decomposed into two trivial arcs embed in two solid tori in a genus one Heegaard splitting of a lens space. A (1,1)-knot can be described by a (1,1)-diagram D(a, b, c, r) determined by four integers a, b, c, and r. It is known that every 2-bride knot is a (1, 1)-knot and has a (1, 1)-diagram of the form D(a, 0, 1, r). In this talk, we give the dual diagram of D(a, 0, 1, r) explicitly and present how to derive a Schubert normal form of a 2-bridge knot from the dual diagram. This gives an alternative proof of the Grasselli and Mulazzani’s result asserting that D(a, 0, 1, r) is a (1, 1)-diagram of 2-bridge knot with a Schubert normal form b(2a+1, 2r).

October 25, 2019

Speaker: Yubin Shin

Title: Image segmentation using persistence and watershed algorithm 

Abstract

Watershed algorithm is one of the segmentation methods, which has a tendency to overdo the segmentation. We correct for this excessive segmentation by distinguishing the desired features from noise by persistent homology from topological data analysis. We will discuss properties of the method including some advantages, disadvantages and algorithmic efficiency.

October 18, 2019

Speaker: Paul Duncan

Title: Cocycle Counting in Random Simplicial Complexes  

Abstract

A rich theory of random graphs has developed since the introduction of the Erdos-Renyi model. Similar questions can be asked of random simplicial complexes, which are much less well understood. I will talk about the technique used in determining thresholds in these complexes for the vanishing of homology, which is analogous to graph connectivity.

October 4, 2020 (Location - MW 154)

Speaker: Josiah Oh

Title: An Introduction to Large-Scale Geometry

Abstract

I plan to give a general survey of some important concepts in geometric group theory, and more specifically, large-scale geometry. I plan to introduce quasi-isometries and their invariants (Gromov hyperbolicity, growth of groups, ends of groups, etc.), and I will state some of the significant theorems that concern these notions. The talk will focus more on breadth than depth, and thus is geared more towards those who are new to (but interested in!) geometric group theory.

September 20,2019

Speaker: Hao Xing

Title: Johnson–Lindenstrauss lemma 

Abstract

The Johnson–Lindenstrauss lemma is a result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. In this short talk, I will give a brief introduction to the Johnson–Lindenstrauss lemma and sketch the idea of the proof.   

September 13, 2019

Speaker: Zhengchao Wan

Title: Gromov-Hausdorff distance on Ultrametric spaces

Abstract

I will introduce a generalized Gromov-Hausdorff distance uGH on the collection U of ultrametric spaces which makes (U,uGH) itself an ultrametric space. This distance function can be characterized by distortion of correspondence as in the case of dGH. Moreover, I will show a structural theorem of uGH that allows us to devise a poly time algorithm to compute uGH.

April 19, 2019 

Speaker: Anastasios Stefanou

Title: Applied topology for phylogenetic networks

Abstract

In mathematical phylogenetics we are interested in developing mathematical methods that model the structure of phylogenetic networks, which in turn, can be used for developing metrics so that any pair of phylogenetic networks can be compared. In practice we often encounter tree-like networks called phylogenetic trees whose structure is already very well understood: one can show (Sokal and Rohlf, 1962) that, via the so called cophenetic map (which will be defined and explained during the lecture), phylogenetic trees with n leaves embed as vectors in the [n(n+1)/2]-dimensional Euclidean space. There we pull back l^p norms to define metrics on these trees. However very few are known for the general case of phylogenetic networks. In this talk I will explain why Reeb graphs is a natural model for phylogenetic networks and I will discuss a canonical way to decompose a phylogenetic network with n-labelled leaves and s cycles into a set of phylogenetic trees with (n + s)-labelled leaves. By combining this tree-decomposition and the cophenetic map, any such network embed as a finite set of points in the [(n+s)(n+s+1)/2]-dimensional Euclidean space. Hence, we can utilize the Hausdorff metric as a metric for comparison of phylogenetic networks.

March 29, 2019

Speaker: Alexander Wagner 

Title: Discrete Morse Theory for Approximating Persistence Diagrams

Abstract

In this talk, I will briefly introduce discrete Morse theory and its extension to the filtered setting before sketching some work-in-progress that aims to use a discrete Morse function on a complex X to approximate the persistent homology of a filtration on X.

March 22, 2019

Speaker: Kathryn Hess

Title: Topological characterization of neuron morphologies

Abstract

The Topological Morphology Descriptor (TMD) encodes complex branching patterns, such as those of neuronal trees, as barcodes. It provides an unbiased benchmark test for the categorization of neuronal morphologies, enabling us to quantify and characterize the structural differences between distinct morphological classes and thus to increase our understanding of the anatomy and diversity of branching morphologies.  We showed that applying the TMD to dendritic arbors of rat PCs  provides an objective, reliable classification into 17 type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or form a morphological continuum.  Applying the TMD to the apical dendrites of  60 3D reconstructed pyramidal neurons from layers 2 and 3 in the human temporal cortex revealed the existence of two morphologically distinct classes that also had distinct electrical behavior.

Work on this project has been led by Lida Kanari.

March 1, 2019 (double talk)

Speaker: Tatsuya Mikami

Title: Percolation on homology generators in codimension one

Abstract

Percolation theory is a branch of probability theory which mainly studies the behavior of clusters in a random graph. Recently, craze formation in polymer materials is gaining attention as a new type of percolation phenomenon in the sense that a large void corresponding to a craze of the polymer starts to appear by the process of coalescence of many small voids. In this talk, I will introduce a new percolation model motivated from the craze formation. For the sake of modeling the coalescence of nanovoids, this model focuses on clusters of "holes" of a random figure, which are formulated as homology generators in codimension one, while the classical percolation theory mainly studies clusters of vertices (i.e., 0-dimensional objects). This is a joint work with Yasuaki Hiraoka. (Reference: https://arxiv.org/abs/1809.07490)

Speaker: Shu Kanazawa

Title: Local structures of random simplicial complexes

Abstract

The notion of local weak convergence of finite graphs, introduced by Benjamini and Schramm in 2001, provided new asymptotic results for random graphs. In recent years, many tools in this framework have turned out to be extremely powerful even for random simplicial complexes. In this talk we will briefly describe the notion of local weak convergence of random simplicial complexes and how to read off the asymptotic behavior of the Betti numbers (nonlocal parameters) from its limiting local structures. Finally, we will report a recent asymptotic result of Betti numbers for a class of random simplicial complexes. 

February 22, 2019

Speaker: Woojin Kim

Title: Multiparameter Persistent Homology for Time-Varying Metric Data

Abstract

Characterizing the dynamics of time-evolving data within the framework of topological data analysis has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). We will discuss (1) how to induce a multiparameter persistent homology as an invariant of a DMS, and (2) stability of these invariants. In order to address the stability, we extend the Gromov-Hausdorff distance on metric spaces to the setting of DMSs. In our framework, the celebrated Bottleneck-GH stability theorem for (static) metric spaces becomes a special case. This is a joint work with Facundo Mémoli. (Reference: https://arxiv.org/abs/1812.00949)

February 15, 2019

Speaker: Fedor Manin

Title: Maps between spheres are hard to optimize

Abstract

It's not too hard to imagine most possible metrics on the 2-sphere.  But higher-dimensional spheres are much weirder.  There are a number of different ways of measuring this, but I will focus on a measure of "roundness" called the hypersphericity.  It turns out that in dimensions 3 and up, the hypersphericity of a given metric sphere is computationally hard to figure out, even approximately. In other words, it's hard to tell a pretty round sphere from a very skinny one "just by looking". This is joint work with Zarathustra Brady and Larry Guth.

November 30, 2018

Speaker: Jimin Kim

Title: Configuration space of thick particles on a metric graph

Abstract:

Configuration space of thick particles on a metric graph is an interesting topic that lies in the intersection of topology, geometry, and combinatorics. Still, it has been little known since Deeley first studied this space in 2011. We will see an example of a configuration space of thick particles on a Y- shaped graph and discuss some of future research problems.

November 16, 2018

Speaker: Woojin Kim

Title: Rank for arbitrary diagrams

Abstract:

The rank of a linear map f is the dimension of the image of f. Hence, the rank of f measures the "nondegenerateness" of f. We extend this notion and define the rank of a diagram of vector spaces and linear maps. We do this, based on a category theoretical perspective. If time permits, we also explore how this generalization can be used in the realm of topological data analysis. This is a joint work with Facundo Memoli.

November 9, 2018

Speaker: Nate Clause

Title: An Approach to Constructing Stable Filtration Functors via Basepoint Filtrations

Abstract:

Persistent Homology is the primary tool used in Topological Data Analysis. Persistent Homology takes a dataset viewed as a finite metric space and converts it into persistence diagrams or barcodes which convey topological information of the underlying dataset. The first step in this process is to use a map called a filtration on the finite metric space, which builds up a simplicial complex on the metric space over time. Currently, a small number of filtrations such as Vietoris-Rips are well studied and put in practice, due to possessing some desirable theoretical properties and having clear geometric and topological intuition. However, there has been no generalized approach for constructing new filtrations which might provide novel insights for datasets. We will define a new method for constructing filtrations via curvature sets and valuations, and then generalize this approach to a method for constructing what we call basepoint filtrations. We outline desirable properties of filtrations constructed through this process, and provide computational examples.

October 26, 2018

Speaker: Francisco Martinez Figueroa

Title: Kneser-Lovasz Theorem

Abstract:

In 1978 Lovasz proved Kneser’s conjecture about the chromatic number of Kneser Graphs. His proof is among the firsts and most prominent examples of an application of algebraic topology tools to a problem about finite combinatorial objects. Many more proofs of this result are now known, most of them involve some topological tools. One of the simplest known proofs is due to Greene (2002) which follows from Borsuk-Ulam’s theorem. We will discuss some properties of Kneser’s graphs and Greene’s proof. 

October 19, 2018

Speaker: Paul Duncan

Title: Percolation

Abstract:

Percolation theory is a study of natural physical models that has drawn considerable attention in the past sixty years. In two dimensions the classical model on the integer lattice is well understood, but there are many possible generalizations to higher dimensions. We will review known results and discuss some topologically motivated ways forward.

September 21, 2018

Speaker: Samir Chowdhury

Title: Computational Optimal Transport

Abstract:

In the past two decades, computational optimal transport has generated significant interest in the computer vision and machine learning communities. We will outline the OT approach to shape matching tasks as described by Rubner, Tomasi, and Guibas (2000), and explain the associated computational problem. Time permitting, we will briefly describe the notion of entropy regularized optimal transport that has been popularized over the past five years.