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Homological Algebra for Persistence ModulesMay 14 2019We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf ... More
Embeddings of Persistence Diagrams into Hilbert SpacesMay 11 2019Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence ... More
Graded persistence diagrams and persistence landscapesApr 29 2019We introduce a refinement of the persistence diagram, the graded persistence diagram. It is a sequence of diagrams whose sum is the persistence diagram. The points in the k-th graded persistence diagram are signed and are the local maxima and minima, ... More
Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in VideosApr 23 2019Apr 24 2019This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the ... More
Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in VideosApr 23 2019This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the ... More
Cotorsion torsion triples and the representation theory of filtered hierarchical clusteringApr 15 2019We give a full classification of representation types of the subcategories of representations of an $m \times n$ rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of clustering ... More
Metrics and stabilization in one parameter persistenceApr 05 2019We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector ... More
Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence ModuleFeb 20 2019May 15 2019A recent work by Lesnick and Wright proposed a visualisation of $2$D persistence modules by using their restrictions onto lines, giving a family of $1$D persistence modules. We give a constructive proof that any $1$D persistence module with finite support ... More
Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangementsFeb 18 2019Feb 26 2019In this work, we explain how to use computational topology for detecting differences in the geometrical distribution of cells forming epithelial tissues. In particular, we extract topological information from images using persistent homology and summarize ... More
Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangementsFeb 18 2019Feb 27 2019In this work, we explain how to use computational topology for detecting differences in the geometrical distribution of cells forming epithelial tissues. In particular, we extract topological information from images using persistent homology and summarize ... More
Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangementsFeb 18 2019In this work, we explain how to use computational topology for detecting differences in the geometrical distribution of cells forming epithelial tissues. In particular, we extract topological information from images using persistent homology and summarize ... More
Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input ... More
Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent HomologyFeb 15 2019Mar 25 2019Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules \[ ... More
Algebraic cobordism of number fieldsJan 13 2019We compute the motivic homotopy groups of algebraic cobordism over number fields, the motivic homotopy groups of 2-complete algebraic cobordism over the real numbers and rings of $2$-integers and the motivic homotopy groups of mod 2 motivic Morava $K$-theory ... More
Supersymmetric Euclidean Field Theories and K-theoryJan 08 2019We construct spaces of 1-dimensional supersymmetric Euclidean field theories and show that they represent real or complex K-theory. A noteworthy feature of our bordism category is that the identity bordism of a point is connected to intervals of positive ... More
Intersection homology duality and pairings: singular, PL, and sheaf-theoreticDec 27 2018Feb 05 2019We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular ... More
On $p$-adic limits of topological invariantsNov 01 2018The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$ completion of ... More
The Persistent Homology of a Sampled Map: From a Viewpoint of Quiver RepresentationsOct 28 2018The theory of homology induced maps of correspondences proposed by Shaun Harker et al. in 2016 is a powerful tool which allows the retrieval of underlying homological information from sampled maps with noise or defects. In this paper, we redefine induced ... More
The equivariant K-theory of isotropy actionsOct 23 2018We compute the equivariant K-theory with integer coefficients of an equivariantly formal isotropy action, subject to natural hypotheses which cover the three major classes of known examples. The proof proceeds by constructing a map of spectral sequences ... More
Characterising epithelial tissues using persistent entropyOct 13 2018In this paper, we apply persistent entropy, a novel topological statistic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geometric information encoded by \alpha-complexes ... More
Sparse Circular Coordinates via Principal $\mathbb{Z}$-BundlesSep 25 2018We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for ... More
Sparse Circular Coordinates via Principal $\mathbb{Z}$-BundlesSep 25 2018May 05 2019We present in this paper an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, we derive, from a 1-dimensional persistent cohomology computation, explicit formulas for ... More
A Brief History of PersistenceSep 10 2018Oct 01 2018Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas, starting from ... More
Variational and Quasi-Variational Inequalities with Gradient Type ConstraintsSep 06 2018This survey on stationary and evolutionary problems with gradient constraints is based on developments of monotonicity and compactness methods applied to large classes of scalar and vectorial solutions to variational and quasi-variational inequalities. ... More
A two species hyperbolic-parabolic model of tissue growthSep 06 2018Models of tissue growth are now well established, in particular in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient generated by the death and birth of cells, itself controlled ... More
The continuity of Darboux injections between manifoldsSep 02 2018Sep 12 2018We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ ... More
Counting maximally broken Morse trajectories on aspherical manifoldsAug 27 2018Jan 24 2019We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.
Geometry of the Phase Retrieval ProblemAug 23 2018One of the most powerful approaches to imaging at the nanometer or subnanometer length scale is coherent diffraction imaging using X-ray sources. For amorphous (non-crystalline) samples, the raw data can be interpreted as the modulus of the continuous ... More
Quantitative Tamarkin categoryJul 25 2018This is a lecture note from a seminar course given at Tel Aviv University in Spring 2018. Part of the preliminary section is built from Kazhdan's seminar organized in the Hebrew University of Jerusalem in Fall 2017. The main topic of this note is a detailed ... More
Magnitude cohomologyJul 18 2018Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we equip with the ... More
Interpolating factorizations for acyclic Donaldson--Thomas invariantsJul 05 2018Mar 04 2019We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rimanyi by counting codimensions of quiver loci, gives two extremal ... More
Interpolating factorizations for acyclic Donaldson--Thomas invariantsJul 05 2018Jul 15 2018We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rim\'anyi by counting codimensions of quiver loci, gives two extremal ... More
Exact computation of the $2+1$ convex hull of a finite setJun 21 2018We present an algorithm to exactly calculate the $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hull of a finite set of points in $\mathbb{R}^3$, as introduced in \cite{KirchheimMullerSverak2003}. When $\mathbb{R}^3$ is considered as certain subset ... More
Machine-learning inference of fluid variables from data using reservoir computingMay 23 2018Aug 10 2018We infer both microscopic and macroscopic behaviors of a three-dimensional chaotic fluid flow using reservoir computing. In our procedure of the inference, we assume no prior knowledge of a physical process of a fluid flow except that its behavior is ... More
Assembly MapsMay 01 2018Jan 02 2019We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of ... More
Piecewise linear sheavesApr 30 2018Nov 28 2018On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then ... More
The Cohomology Algebra of Polyhedral Product ObjectsApr 21 2018In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into commutative algebra ... More
Discrete Cubical and Path Homologies of GraphsMar 20 2018In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Caprano, and White, is based on graph maps from hypercubes to the graph. The second theory was developed by Grigor'yan, ... More
Re-examination of Bregman functions and new properties of their divergencesMar 01 2018Apr 08 2019The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during ... More
Re-examination of Bregman functions and new properties of their divergencesMar 01 2018Nov 20 2018The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during ... More
Topological spaces of persistence modules and their propertiesFeb 22 2018Oct 25 2018Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including ... More
Higher Hochschild homology and exponential functorsFeb 21 2018Jun 14 2018We study higher Hochschild homology evaluated on wedges of circles, viewed as a functor on the category of free groups. The principal results use coefficients arising from square-zero extensions; this is motivated in part by work of Turchin and Willwacher ... More
Algebraic Intersection SpacesFeb 12 2018Dec 05 2018We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a ... More
On the geometrical properties of the coherent matching distance in 2D persistent homologyJan 20 2018May 25 2018In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including ... More
Local Coordinate Spaces: a proposed unification of manifolds and fiber bundles, and associated machineryJan 17 2018This paper presents a unified view of manifolds and fiber bundles, which, while superficially different, have strong parallels. It introduces the notions of an m-atlas and of a local coordinate space, and shows that special cases are equivalent to fiber ... More
On the Structure of Algebraic CobordismDec 11 2017May 31 2018In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with ... More
Discrete Morse-Bott theory for CW complexesNov 29 2017We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory. ... More
Four-dimensional graph-manifolds with fundamental groups quasi-isometric to fundamental groups of orthogonal graph-manifoldsNov 23 2017We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover ... More
A Case Study in Non-Commutative TopologyNov 23 2017This is an expository note focused upon one example, the irrational rotation $C^*$-algebra. We discuss how this algebra arises in nature - in quantum mechanics, group actions, and foliations, and we explain how $K$-theory is used to get information out ... More
Catalan States of Lattice Crossing: Application of Plucking PolynomialNov 14 2017For a Catalan state $C$ of a lattice crossing $L\left( m,n\right) $ with no returns on one side, we find its coefficient $C\left( A\right) $ in the Relative Kauffman Bracket Skein Module expansion of $L\left( m,n\right) $. We show, in particular, that ... More
Symmetric groups and checker triangulated surfacesOct 31 2017We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of permutations ... More
Generalized information structures and their cohomologySep 22 2017Oct 12 2018D. Bennequin and P. Baudot introduced a cohomological construction adapted to information theory, called 'information cohomology', that characterizes Shannon entropy through a cocycle condition. This text develops the relation between information cohomology ... More
Weighted Persistent HomologyAug 31 2017Dec 06 2018We introduce weighted versions of the classical \v{C}ech and Vietoris-Rips complexes. We show that a version of the Vietoris-Rips Lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary ... More
Weighted Persistent HomologyAug 22 2017Apr 14 2018In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology ... More
Notes on the bounded cohomology theoryAug 17 2017The paper is devoted to a generalized and improved version of author's approach to Gromov bounded cohomology theory. In particular, the awkward countability assumption is removed and the aspects related to homological algebra are clarified. The exposition ... More
Matrix Method for Persistence Modules on Commutative Ladders of Finite TypeJun 30 2017Sep 25 2018The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view a persistence ... More
Persistent homology and microlocal sheaf theoryMay 02 2017Sep 07 2018We interpret some results of persistent homology and barcodes (in any dimension) with the language of microlocal sheaf theory. For that purpose we study the derived category of sheaves on a real finite-dimensional vector space V. By using the operation ... More
Hilbert-Poincare series for spaces of commuting elements in Lie groupsApr 19 2017Jul 25 2018In this article we study the homology of spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. ... More
Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution AnalysisMar 31 2017May 29 2017We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing ... More
On the set of optimal homeomorphisms for the natural pseudo-distance associated with the Lie group S^1Mar 04 2017If $\varphi$ and $\psi$ are two continuous real-valued functions defined on a compact topological space $X$ and $G$ is a subgroup of the group of all homeomorphisms of $X$ onto itself, the natural pseudo-distance $d_G(\varphi,\psi)$ is defined as the ... More
Twisted monodromy homomorphisms and Massey productsJan 23 2017Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\rho:\pi_1(X)\to GL(n,\mathbb{C})$ ... More
Homology and homotopy complexity in negative curvatureDec 14 2016Oct 04 2018Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of ... More
Integral chains and Bousfield-Kan completionNov 13 2016Oct 12 2018Working in the Arone-Ching framework for homotopical descent, it follows that the Bousfield-Kan completion map with respect to integral homology is the unit of a derived adjunction. We prove that this derived adjunction, comparing spaces with coalgebra ... More
The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applicationsNov 02 2016Jun 27 2017We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological features of a compact ... More
The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applicationsNov 02 2016A persistent diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative ... More
The first Cheeger constant of a simplexOct 23 2016The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial complex $X$, and any $k\geq ... More
The first Cheeger constant of a simplexOct 23 2016Sep 05 2017The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial complex $X$, and any $k\geq ... More
An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domainOct 07 2016In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell's equations ... More
Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distancesOct 05 2016We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a variety of tools ... More
Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distancesOct 05 2016Apr 28 2017We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a variety of tools ... More
Massey products in mapping toriOct 04 2016Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\lambda\in \mathbb{Z}^*$. ... More
Massey products in mapping toriOct 04 2016Dec 20 2016Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\lambda\in \mathbb{Z}^*$. ... More
Some results on the topology of real Bott towersSep 19 2016The main aim of this article is to study the topology of real Bott towers as special and interesting examples of real toric varieties. We first give a presentation of the fundamental group of a real Bott tower and show that the fundamental group is abelian ... More
Asymptotic observables in gapped quantum spin systemsAug 31 2016This paper gives a construction of certain asymptotic observables (Araki-Haag detectors) in ground state representations of gapped quantum spin systems. The construction is based on general assumptions which are satisfied e.g. in the Ising model in strong ... More
Parabolic Kazhdan-Lusztig basis, Schubert classes, and equivariant oriented cohomologyAug 23 2016We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can ... More
Tight framelets and fast framelet filter bank transforms on manifoldsAug 13 2016Mar 01 2018Tight framelets on a smooth and compact Riemannian manifold $\mathcal{M}$ provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications ... More
Tight framelets and fast framelet transforms on manifoldsAug 13 2016Tight framelets on a smooth and compact Riemannian manifold $\mathcal{M}$ provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications ... More
Classifying conformally invariant loop measuresAug 13 2016We formulate a classification conjecture for conformally invariant families of measures on simple loops that builds on a conjecture of Kontsevich and Suhov. The main example in this class of objects was constructed by Werner as boundaries of Brownian ... More
On the Surjectivity of Certain MapsAug 12 2016Nov 16 2016We prove in this article the surjectivity of three maps. We prove in Theorem $1$ the surjectivity of the chinese remainder reduction map associated to projective space of an ideal with a given factorization into ideals whose radicals are pairwise distinct ... More
The Morse-Bott inequalities, orientations, and the Thom isomorphism in Morse homologyJul 21 2016The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions. We show that ... More
Higher cohomology operations and R-completionJul 11 2016Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$ up to $R$-completion. ... More
Higher cohomology operations and R-completionJul 11 2016Dec 11 2017Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$ up to $R$-completion. ... More
Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficientsJun 24 2016Jun 30 2017Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application ... More
Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficientsJun 24 2016Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application ... More
A remark on the convergence of Betti numbers in the thermodynamic regimeJun 21 2016The convergence of the expectations of Betti numbers of \v{C}ech complexes built on binomial point processes in the thermodynamic regime is established.
A new class of homology and cohomology 3-manifoldsJun 19 2016Jun 26 2016We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology 3-manifold with coefficients ... More
Is a monotone union of contractible open sets contractible?Jun 16 2016Oct 10 2016This paper presents some partial answers to the following question. QUESTION. If a normal space X is the union of an increasing sequence of open sets U(1), U(2), U(3) ... such that each U(n) contracts to a point in X, must X be contractible? The main ... More
Platonic and alternatinc 2-groupsMay 30 2016We recall Schur's work on universal central extensions and develop the analogous theory for categorical extensions of groups. We prove that the String 2-groups are universal in this sense and study in detail their restrictions to the finite subgroups ... More
New invariants for a real valued and angle valued map (an Alternative to Morse- Novikov theory)May 24 2016This paper but section 6 is essentially my lecture at The Eighth Congress of Romanian Mathematicians, June 26 - July 1, 2015, Iasi, Romania. The paper summarizes the definitions and the properties of the invariants associated to a real or an angle valued ... More
Higher interpolation and extension for persistence modulesMar 24 2016Mar 20 2017The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion ... More
Position paper: Towards an observer-oriented theory of shape comparisonMar 07 2016In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions ... More
A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)Mar 06 2016Mar 25 2016This paper is a sequel of [2]. We propose a refinement of the Novikov-Betti numbers and of the Novikov homology (w.r. to a specified field) of a pair consisting of a compact ANR and a degree one integral cohomology class in the presence of a continuous ... More
Equivariant A-infinity algebras for nonorientable LagrangiansDec 14 2015Jan 28 2016We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted $A_\infty$ algebras ... More
Stabilizing the unstable output of persistent homology computationsDec 05 2015Oct 25 2018We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ... More
Logic of differential calculus and the zoo of geometric strujcturesNov 21 2015Since the discovery of differential calculus by Newton and Leibniz and the subsequent continuous growth of its applications to physics, mechanics, geometry, etc, it was observed that partial derivatives in the study of various natural problems are (self-)organized ... More
Topological Rigidity ProblemsOct 11 2015Oct 15 2015We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during ... More
Bivariant Versions of Algebraic CobordismSep 02 2015We define four distinct oriented bivariant theories associated with algebraic cobordism in its two versions (the axiomatic $\Omega$ and the geometric $\omega$), when applied to quasi-projective varieties over a field $k$. Specifically, we obtain contravariant ... More
A Schubert basis in equivariant elliptic cohomologyAug 13 2015Feb 06 2016We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two ... More
Hochschild-Pirashvili homology on suspensions and representations of $Out(F_n)$Jul 30 2015Apr 02 2016We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f\colon \Sigma Y\to \Sigma Z$, the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ ... More
Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and ColloidsJul 06 2015Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent homology) rank ... More