# Poster session

Tuesday, May 22, 2018 - 2:30pm - 3:15pm

Lind 400

**Certified Mapper: Obstructions and good covers**

Mikael Vejdemo-Johansson (City University of New York (CUNY))

The Mapper construction comes close to producing a good cover in the sense of the nerve lemma, but by using $\pi_0$ on cover elements it brings no guarantees of actually producing a good cover. For a homological nerve lemma, we can use local persistent homology calculations to seek homological obstructions, or a certificate of how well a Mapper cover matches with the nerve lemma claim. We describe two different ways to achieve such a certificate: one using a persistent nerve lemma, and one using simulation testing and numeric barcode invariants.**Stability of the Leray spectral sequence**

Dustin Sauriol (Colorado State University)

Spectral sequences are powerful tools in homological algebra used to compute homology groups in a successive manner. This is advantageous as it often allows additional information to be deduced beyond the homology group calculation. In particular, the Leray spectral sequence gives information on how the fibers of a fibration piece together over a base space and gives a sense of how complicated the map is. Stability of these sequences is crucial in order to be suitable for use in any quantitative topology setting such as data analysis. Here we examine the stability of the Leray spectral sequence and explore some immediate consequences.**Sheaf-theoretic stratification learning**

Adam Brown (The University of Utah)

We will briefly investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Rourke and Sanderson on homological stratifications, we outline a stratification learning algorithm framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. This algorithm computes a coarsest stratification for which the given sheaf is constructible. We envision that our sheaf-theoretic algorithm could give rise to a larger class of (computable) stratifications beyond homology-based stratification.**The Diffusion Geometry of Fibre Bundles**

Tingran Gao (University of Chicago)

The algorithm Horizontal Diffusion Maps generalizes the framework of diffusion geometry to fibre bundles equipped with a connection, for modeling datasets with pairwise structural correspondences. The horizontal diffusion is defined on the fibre bundle space but driven by a diffusion process on the base manifold; the infinitesimal generator of this novel diffusion process is the horizontal Laplacian in the context of Riemannian submersion. We demonstrate the efficacy of this computational framework in real biological shape classification problems in evolutionary anthropology.**The Mayer-Vietoris Pyramid Sheaf-theoretically**

Benedikt Fluhr (Technical University of Munich)

We provide a sheaf-theoretical version of the Mayer-Vietoris pyramid, introduced by Carlsson, de Silva, and Morozov, together with a notion of interleavings. We hope we can use this to gain an understanding how interleavings of levelset persistence introduced by Curry relate to interleavings of extended persistence introduced by Bubenik and Scott. Another hope is we can extend some of our ideas to situations where only partial information is available. Our constructions are heavily inspired by Happel's descriptions of the derived categories of Dynkin quivers and the Abelianization of triangulated categories.**Homological Algebra of Persistence Modules**

Nikola Milicevic (University of Florida)

Persistence modules play a central role in topological data analysis. For example they can be obtained from a continuous real valued function on a topological space by computing the homology of the sub-level sets. We consider persistence modules from two points of view, an algebraic one and a category theory one. One can use the the algebraic point of view to define a tensor product of persistence modules. We show the tensor product has a right adjoint functor and we construct a new category of persistence modules that is enriched over itself in the process. As a consequence a new adjunction arises in the original category. Since the original category has the necessary ingredients to do homological algebra we begin to develop this theory using this pair of adjunct functors. More specifically we show that every persistence module has a projective and an injective resolution. We classify the projective and injective interval modules and use them as building blocks of projective and injective resolutions. This is joint work with Peter Bubenik.**Statistical Tools for Persistent Homology Using Optimal Transport**

Théo Lacombe (INRIA Saclay - Île-de-France )

The space of persistence diagrams is naturally endowed with the Bottleneck distance: a partial matching metric deduced from the interleaving distance between persistence modules, leading to strong stability results with respect to the input data. However, the combinatorial nature of the bottleneck distance makes it unhandy for a statistical purpose: it is numerically hard to compute and to optimize with. Thus, many basic statistical tools as simple as barycenters are intractable when dealing with persistence diagrams. We present in this poster a relaxation of diagram distances that circumvents these issues by leveraging recent progress in computational optimal transport.**A Model for Random Chain Complexes**

Michael Catanzaro (University of Florida)Matthew Zabka (Southwest Minnesota State University)

Topology's tools have become popular for analyzing data, and this has raised questions as to how randomness can be added to topological ideas. In this poster, we shall give a model for a random, finite-dimensional, graded chain complex over a finite field, whose randomness is given by the entries of the matrices that define the boundary operators. We then investigate some the random complex’s probabilistic properties.**A derived isometry theorem for sheaves**

Nicolas Berkouk (INRIA )

Following Curry's, Kashiwara and Schapira's work on adapting persistence theory ideas to sheaves on a real vector space in the derived setting, we extensively study their so-called convolution distance on the category of constructible sheaves over the real line $\D^b_{\R c}(\kk_\R)$.

In this setting, sheaves naturally possess a notion of graded-barcode. We propose in particular to define a bottleneck distance between graded-barcodes, and to prove an isometry theorem : the convolution distance between two sheaves is equal to the bottleneck distance between their associated graded barcodes.**Constructible cosheaves over the Ran space**

Janis Lazovskis (University of Illinois, Chicago)

Every finite collection of points on a manifold has a simplicial complex associated to it, the Cech complex, once a positive radius is chosen. The space of all such collections and all such radii admits a cosheaf valued in simplicial complexes whose stalk at every point is this associated simplicial complex. Over larger open sets we take the colimit over all entry paths that respect the partition of the base space by the associated simplicial complex. The stratification involved in this construction is not conical, however when restricted to piecewise-linear manifolds, the existence of a compatible conical stratification is guaranteed by semialgebraic geometry.