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Brittany Fasy
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Michael Lesnick
The stability theorem for persistent homology is a central result in topological data analysis.
While the original formulation of the result concerns the persistent homology of mathbb{R}-
valued functions, the result was later cast in a more general algebraic form, in the language of
persistence modules and interleavings. In this work, we establish an analogue of this algebraic
stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag
persistence module to a two-dimensional persistence module, and establish an algebraic stability
theorem for these extensions. As an application of our main theorem, we strengthen a result of
Bauer, Munch, and Wang on the stability of the persistent homology of Reeb graphs. Our main
result also yields an alternative proof of the stability theorem for level set persistent homology of
Carlsson et al.
This is joint work with Magnus Botnan.
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Amit Patel
The persistence diagram is very different in philosophy from the barcode. Suppose we have a constructible persistence module of vector spaces. Its barcode is its list of indecomposables. Its persistence diagram is an encoding of all persistent vector spaces. In the setting of vector spaces, we know that these two notions are equivalent. However, we quickly run into problems if we try to generalize the barcode beyond the setting of vector spaces. In this talk, I will generalize the persistence diagram to the setting of constructible persistence modules valued in any symmetric monoidal category. For example, the category of sets, the category of vector spaces, and the category of abelian groups are symmetric monoidal categories. As an immediate consequence, we can finally talk about persistent homology over integer coefficients!
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Herbert Edelsbrunner
Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in R^n, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we study general properties of the discrete gradient and obtain precise expressions for the expected numbers in low dimensions. In particular, we get the expected numbers of simplices in the Poisson--Delaunay mosaic in dimension n <= 4.
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Jose Perea
One of the main challenges in topological data analysis is to turn computed topological features, such as barcodes, into insights about the data set under analysis.
We will show in this talk how the persistent cohomology of sparse Cech filtrations (introduced recently by D. Sheehy et. al.), in dimensions 1 and 2, can be used to construct robust representations of the data in the real and complex projective spaces. Examples will be presented in order to illustrate how projective coordinates provides a framework for topology-driven nonlinear dimensionality reduction, and geometric model generation.
This work extends results of V. de Silva, D. Morozov and M. Vejdemo-Johansson on persistent cohomology and circular coordinates.
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Vin de Silva
I shall talk about Reeb graphs, and the benefits of regarding them as set-valued cosheaves on the real line. For instance, there is a smoothing operation and an interleaving distance for these cosheaves, which can be interpreted as a smoothing operation and an interleaving distance for Reeb graphs. This is joint work with Elizabeth Munch and Amit Patel. If time allows, I will discuss various extensions of these ideas, worked out with the additional collaboration of Anastasios Stefanou; and also of Pomona College undergraduates Song Yu and Dmitriy Smirnov.
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Bei Wang
The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper (Singh et al.), and a special case of the mapper construction called the Joint Contour Net (Carr et al. ) have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit.
We are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.
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Justin Curry
In this talk I will introduce the realization problem in persistence, which asks what isomorphism classes of diagrams of vector spaces can be realized by diagrams of topological spaces and continuous maps. In particular, one could ask what barcodes can be obtained by filtering a simplicial complex or by studying the level-set persistence of a map. I will review existing results for point clouds, and present some results of my own, obtained in collaboration with Ulrich Bauer and Hans Reiss. These results include studying the level-set barcodes realized by a stratified space and real-valued map, the space of barcodes realized by filtering a manifold by Gauss curvature, and the space of barcodes realized by Morse functions. To address which barcodes can be realized as the level-set barcodes of a Morse function $f$, I will present two constructions. One construction uses handlebody theory. The other construction is more novel and uses a cosheaf of spaces over the Reeb graph of $f$, which incidentally makes headway into a problem posed by Arnold. Additionally, this construction offers a vision for extending Mapper degree by degree (in analogy with Postnikov towers), offering a potentially powerful new tool in topological data analysis.
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Jeff Erickson
Any generic closed curve in the plane can be transformed into a simple closed curve by
a finite sequence of local transformations called homotopy moves. We prove that simplifying a
planar curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case.
The best bounds previously known were a 100-year-old O(n^2) upper bound due to Steinitz and
the trivial Omega(n) lower bound. Our lower bound also implies that Omega(n^{3/2}) degree-1
reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any
planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for
rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are
required in the worst case to transform one non-contractible closed curve on the torus to another;
this lower bound is tight if the curve is homotopic to an embedding.
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Dan Burghelea
It is well known how basic Algebraic Topology and geometrization of Large Data led
to Persistence Theory, a useful tool in Data Analysis.
In this talk I will explore the other direction; how Persistence Theory suggests and motivates
refinements of some basic topological invariants, like homology and Betti numbers, and suggests
alternative descriptions of others invariants, like monodromy, of mathematical relevance and
with computational implications. The mathematics described is a part of what I refer to as an
ALTERNATIVE to MORSE-NOVIKOV theory.
The refinements proposed are in terms of configurations of vector spaces for the relevant
homologies, and in terms of polynomials for Betti numbers. The alternative description of
monodromy is computer friendly, hence without the need of infinite objects (infinite cyclic
cover). A few applications of these refinements in topology, geometric analysis and dynamics
might be indicated.
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Sanjeevi Krishnan
Every connected, finite CW complex is homotopy equivalent to the classifying space
of a monoid [McDuff, 1979], a set with an associative and unital multiplication. Thus group
completions of monoids correspond to fundamental groups of based spaces. We discuss concrete
algorithms for computing (co)homology and higher homotopy as algebraic constructions on
monoids.
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Robert Ghrist
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Gunnar Carlsson
Motivated by work on data sets coming from the biology and evolution, we have some
results concerning the persistent homology of trees and also products of finite metric spaces.
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Elizabeth Munch
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Don Sheehy
In this talk I will discuss techniques and heuristics for subsampling metric data as well as a space
of tree-like data structures that one might build on top of such samples, generalizing cover trees,
net trees, navigating nets, deformable spanners, and some classes of hierarchical spanners.
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Peter Bubenik
Persistence modules are the central algebraic object in topological data analysis. This
motivates the study of the geometry of the space of persistence modules. We isolate an elegant
coherence condition that guarantees the interpolation and extension of sets of persistence
modules. This "higher interpolation" is a consequence of the existence of certain universal
constructions. As an application, it allows one to compare Vietoris-Rips and Cech complexes
built within the space of persistence modules. This is joint work with Vin de Silva and Vidit
Nanda.
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Benjamin Schweinhart
Although random cell complexes occur throughout the physical sciences, there does
not appear to be a standard way to quantify their statistical similarities and differences. I'll
introduce the method of swatches, which describes the local topology of a cell complex in terms
of probability distributions of local configurations. It allows a distance to be defined which
measures the similarity of the local topology of cell complexes. Convergence in this distance is
related to the notion of a Benjamini Schramm graph limit. In my talk, I will use this to state
universality conjectures about the long-term behavior of graphs evolving under curvature flow,
and to test these conjectures computationally. This system is of both mathematical and physical
interest.
If time permits, I will discuss other applications of computationally topology to curvature flow
on graphs, and describe recent work on a new notion of geometric graph limit.
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Vidit Nanda
Large-scale homology computations are often rendered tractable by discrete Morse
theory. Every discrete Morse function on a given cell complex X produces a Morse chain
complex whose chain groups are spanned by critical cells and whose homology is isomorphic to
that of X. However, the space-level information is typically lost because very little is known
about how critical cells are attached to each other. In this talk, we discretize a beautiful
construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X
from an overlaid discrete Morse function.
