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The diagonal of the associahedraFeb 21 2019This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff ... More
Permutations of Integers and pseudo-Anosov mapsFeb 20 2019We will prove that an ordered block permutation (OBP) (a permutation of n positive integers) when admissible, corresponds to an oriented-fixed (OF) pseudo-Anosov homeomorphism of a Riemann surface (with respect to an Abelian differential and fixing all ... More
Approximating Continuous Functions on Persistence Diagrams Using Template FunctionsFeb 19 2019The persistence diagram is an increasingly useful tool arising from the field of Topological Data Analysis. However, using these diagrams in conjunction with machine learning techniques requires some mathematical finesse. The most success to date has ... More
On the $v_1$ periodicity of the Moore spaceFeb 19 2019We present progress in trying to verify a long-standing conjecture by Mark Mahowald on the $v_{1}$-periodic component of the classical Adams spectral sequence for a Moore space $M$. The approach we follow was proposed by John Palmieri in his work on the ... More
Lickorish type construction of manifolds over simple polytopesFeb 19 2019This paper is a survey on the Lickorish type construction of some kind of closed manifolds over simple convex polytopes. Inspired by Lickorish's theorem, we propose a method to describe certain families of manifolds over simple convex polytopes with torus ... More
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More
A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019Feb 19 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More
Morse-Bott Cohomology from Homological Perturbation TheoryFeb 18 2019In this paper, we construct cochain complexes generated by cohomology of critical manifolds for Morse-Bott theory under minimum transversality assumptions. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. ... More
Fluid Modeling and Boolean Algebra for Arbitrarily Complex Topology in Two DimensionsFeb 18 2019We propose a mathematical model for fluids in multiphase flows in order to establish a solid theoretical foundation for the study of their complex topology, large geometric deformations, and topological changes such as merging. Our modeling space consists ... More
Homotopic distance between functorsFeb 17 2019We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors to the context of small categories. Moreover, this notion generalizes the work ... More
Rational homotopy theory for moduli of stable rational curvesFeb 17 2019We show that rational cohomology algebras of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. As a consequence, we compute the rational homotopy algebras of those spaces, and use it to deduce exponential ... More
Double coverings of arrangement complements and $2$-torsion in Milnor fiber homologyFeb 17 2019We prove that mod $2$ Betti numbers of the double covering of a complex hyperplane arrangement complement is combinatorially determined. The proof is based on a relation between mod $2$ Aomoto complex and the transfer long exact sequence. Applying the ... More
Linear motion planning with controlled collisions and pure planar braidsFeb 17 2019We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $s\geq2$) of the "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity ... More
On Leibniz cohomologyFeb 16 2019In this paper we prove the Leibniz analogues of several vanishing theorems for the Chevalley-Eilenberg cohomology of Lie algebras. In particular , we obtain the second Whitehead lemma for Leibniz algebras. Our main tools are three spectral sequences. ... More
Classification of Phylogenetic NetworksFeb 15 2019By considering rooted Reeb graphs as a model for phylogenetic networks, using tools from category theory we construct an injection that assigns to each phylogenetic network with $n$-labelled leaves and $s$ cycles a finite set of phylogenetic trees with ... 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
Nerves of 2-categories and 2-categorification of $(\infty,2)$-categoriesFeb 14 2019We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's model structure ... More
The chromatic Brauer category and its linear representationsFeb 14 2019The Brauer category is a symmetric strict monoidal category that arises as a categorification of the Brauer algebras in the context of Banagl's framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment ... More
A short introduction to the telescope and chromatic splitting conjecturesFeb 13 2019In this note, we give a brief overview of the telescope conjecture and the chromatic splitting conjecture in stable homotopy theory. In particular, we provide a proof of the folklore result that Ravenel's telescope conjecture for all heights combined ... More
Localization for coarse homology theoriesFeb 13 2019We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated ... More
Lax limits of model categoriesFeb 13 2019For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying $\infty$-categories. Our approach can also allow for the ... More
Perverse sheaves on semi-abelian varieties -- a survey of properties and applicationsFeb 13 2019We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various obstructions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), ... More
k-Dismantlability in GraphsFeb 12 2019Given a finite undirected graph $X$, a vertex is $0$-dismantlable if its open neighborhood is a cone and $X$ is $0$-dismantlable if it is reducible to a single vertex by successive deletions of $0$-dismantlable vertices. By an iterative process, a vertex ... More
On defining products of cobordism classes of Morse functionsFeb 11 2019(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch ... More
The topology of Baumslag-Solitar representationsFeb 11 2019Let $\Gamma=\langle a,b | a b^{p} a^{-1} = b^{q}\rangle$ be a Baumslag--Solitar group and $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, ... More
Formal RingsFeb 10 2019A notion of one-dimensional formal ring is presented. It consists of a triple $(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket x,y\rrbracket}$, the first ... More
Principal Postnikov towers and TQ localization of structured ring spectraFeb 09 2019The aim of this paper is to establish that every (-1)-connected algebra over a spectral operad O is TQ-local, in the sense that the natural coaugmentation map to its topological Quillen localization (this can be thought of as the part of the O-algebra ... More
On d-Categories and d-OperadsFeb 09 2019We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization of the d-homotopy ... More
Formality criteria in terms of higher Whitehead bracketsFeb 08 2019We provide two criteria for discarding the formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogue of the Massey products of a differential graded associative algebra. We also show that formality ... More
The Cohomology of the Ordinals I: Basic Theory and Consistency ResultsFeb 07 2019In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is \v{C}ech cohomology. We show in particular that the \v{C}ech cohomology groups of the ordinals articulate ... More
Generalized Dehn twists on surfaces and homology cylindersFeb 07 2019Let $\Sigma$ be a compact oriented surface. The Dehn twist along every simple closed curve $\gamma \subset \Sigma$ induces an automorphism of the fundamental group $\pi$ of $\Sigma$. There are two possible ways to generalize such automorphisms if the ... More
Operads without cogebrasFeb 07 2019We give an example of a non-trivial linear operad that only admits trivial cogebras and give sufficient conditions ensuring that the cofree cogebra functor be faithful.
Digital Hurewicz Theorem and Digital Homology theoryFeb 06 2019In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we present digital Hurewicz theorem for the fundamental group of digital images. We also show that the ... More
Crossed modules and symmetric cohomology of groupsFeb 05 2019This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff ... More
Stability in the cohomology of the space of complex irreducible polynomials in several variablesFeb 05 2019We prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d$ increases, and second, its compactly supported cohomology stabilizes as $n$ ... More
Polynomiality of Grothendieck groups for finite general linear groups, Deligne-Lusztig characters, and injective unstable modulesFeb 05 2019Let K 0 (Fp GLn(Fp)-proj) denote the Grothendieck group of finitely generated pro-jective Fp GLn(Fp)-modules. We show that the algebra C $\otimes$ n$\ge$0 K 0 (Fp GLn(Fp)-proj) with multiplication given by induction functors, is a polynomial algebra. ... More
An optimal Borsuk-Ulam theorem for products of spheres and Stiefel manifoldsFeb 03 2019We give a new, simplified proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in ... More
Hodge cycles for cubic hypersurfacesFeb 03 2019We study an algebraic cycle of the form $Z_0= r {\mathbb P}^{\frac{n}{2}}+\check r \check{\mathbb P}^{\frac{n}{2}}$, $r \in{\mathbb N},\check r \in{\mathbb Z},\ \ 1\leq r , |\check r |\leq 10,\ \ \gcd ( r ,\check r )=1$, inside the cubic Fermat variety ... More
Beyond Sperner's lemmaFeb 03 2019The present paper is devoted to a recent beautiful and ingenious proof of Brouwer's fixed point theorem due to mathematical economists H. Petri and M. Voorneveld. The heart of this proof is an analogue of Sperner's lemma motivated by Shapley-Scarf model ... More
On monomial Golod idealsFeb 02 2019Feb 10 2019We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod.
KO-theory of complex flag varieties of ordinary typeFeb 02 2019We compute the topological Witt groups of every complex flag manifold of ordinary type, and thus the interesting (i.e. torsion) part of the KO-groups of these manifolds. Equivalently, we compute Balmer's Witt groups of each flag variety of ordinary type ... More
Fibrations, unique path lifting, and continuous monodromyJan 31 2019Every covering projection is a fibration with unique path lifting and discrete fibers. In turn, for every fibration with unique path lifting, the monodromies between fibers are continuous and the fundamental group functor solves the unique lifting problem. ... More
An exotic presentation of Q_28Jan 30 2019We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.
Persistent Homology of Geospatial Data: A Case Study with VotingJan 29 2019A crucial step in the analysis of persistent homology is transformation of data into a simplicial complex. Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they ... More
Some Torsion Classes in the Chow ring and Cohomology of $BPGL_n$Jan 29 2019In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. Similarly, ... More
Stable resolutions of multi-parameter persistence modulesJan 28 2019The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties. ... More
Random Simplicial Complexes, Duality and The Critical DimensionJan 28 2019In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the Betti numbers ... More
The Segal Conjecture for Infinite GroupsJan 26 2019We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model for the classifying ... More
Floer homotopy theory, revisitedJan 24 2019In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is ... More
Higher Whitehead products in moment-angle complexes and substitution of simplicial complexesJan 23 2019We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-complex $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated ... More
P-Toral Approximations Compute Bredon HomologyJan 22 2019We study Bredon homology approximations for spaces with an action of a compact Lie group G. We show that if M is a coMackey functor satisfying mild p-locality conditions, then Bredon homology of a G-space X with coefficients in M is determined by fixed ... More
On the homotopy type of complexes of graphs with bounded domination numberJan 22 2019Let $D_{n,\gamma}$ be the complex of graphs on $n$ vertices and domination number at least $\gamma$. We prove that $D_{n,n-2}$ has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of ... More
Generalised BGP reflection functors via the Grothendieck constructionJan 21 2019Feb 07 2019Inspired by work of Ladkani, we explain how to construct generalisations of the classical reflection functors of Bern\v{s}te\u{\i}n, Gel'fand and Ponomarev by means of the Grothendieck construction.
The Lie algebra associated with the lower central series of a right-angled Coxeter groupJan 21 2019We study the lower central series of a right-angled Coxeter group $RC_K$ and the associated Lie algebra $L(RC_K)$. The latter is related to the graph Lie algebra $L_K$. We give an explicit combinatorial description of the first three consecutive factors ... More
Partial Torelli groups and homological stabilityJan 20 2019We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving ... More
$\mathbb{E}_\infty$ automorphisms of motivic Morava $E$-theoriesJan 17 2019We apply Goerss--Hopkins obstruction theory for motivic spectra to study the motivic Morava $E$-theories. We find that they always admit $\mathbb{E}_\infty$ structures, but that these may admit "exotic" $\mathbb{E}_\infty$ automorphisms not coming from ... More
Euclidean distance degree of projective varietiesJan 16 2019Jan 28 2019We give a positive answer to a conjecture of Aluffi-Harris on the computation of the Euclidean distance degree of a possibly singular projective variety in terms of the local Euler obstruction function.
Invertible phases of matter with spatial symmetryJan 16 2019We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, ... More
On transfer maps in the algebraic $K$-theory of spacesJan 16 2019We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to \emph{weak} homotopy with respect to transfer maps in algebraic ... More
Spaces of directed paths on pre-cubical sets IIJan 16 2019For a given pre-cubical set ($\square$--set) $K$ with two distinguished vertices $\bO$, $\bI$, we prove that the space $\vP(K)_\bO^\bI$ of d-paths on the geometric realization of $K$ with source $\bO$ and target $\bI$ is homotopy equivalent to its subspace ... More
The Little Bundles OperadJan 15 2019Feb 10 2019Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little ... More
Categories internal to crossed modulesJan 15 2019Jan 25 2019In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares, and the category ... More
Definable coaisles over rings of weak global dimension at most oneJan 14 2019In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair ... More
The Real Graded Brauer groupJan 14 2019We introduce a version of the Brauer--Wall group for Real vector bundles of algebras (in the sense of Atiyah), and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of ... More
Distributed Monitoring of Topological Events via HomologyJan 14 2019Topological event detection allows for the distributed computation of homology by focusing on local changes occurring in a network over time. In this paper, a model for the monitoring of topological events in dynamically changing regions will be developed. ... 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
K-theory and immersions of spatial polygon spacesJan 13 2019For ell a generic n-tuple of positive numbers, N(ell) denotes the space of isometry classes of oriented n-gons in R^3 with side lengths specified by ell. We determine the algebra K(N(ell)) and use this to obtain nonimmersions of the 2(n-3)-manifold N(ell) ... More
Localizing the $E_2$ page of the Adams spectral sequenceJan 12 2019There is only one nontrivial localization of $\pi_*S_{(p)}$ (the chromatic localization at $v_0=p$), but there are infinitely many nontrivial localizations of the Adams $E_2$ page for the sphere. The first non-nilpotent element in the $E_2$ page after ... More
Comparison of Waldhausen constructionsJan 11 2019In previous work, we develop a generalized Waldhausen $S_{\bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{\bullet}$-constructions ... More
Cup product on $A_\infty$-cohomology and deformationsJan 11 2019Feb 01 2019We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_\infty$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on the $A_\infty$-cohomology, ... 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
Geometry of compact lifting spacesJan 07 2019We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in a previous paper (Conner-Herfort-Pavesic: Some anomalous ... More
On the cohomology of Torelli groupsJan 07 2019We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^g S^n \times S^n$ relative to a disc in a stable range, for $2n \geq 6$. Our calculation is also valid for $2n=2$ assuming that the rational ... More
Cohomology jump loci of 3-manifoldsJan 05 2019The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ... More
On the various notions of Poincaré duality spaceJan 01 2019Jan 14 2019We prove some foundational results on Poincar\'e spaces, which result in two applications. The first is a solution to a conjecture of C.T.C. Wall. The other application is a relative version of a result of Gottlieb about fibrations and Poincar\'e duality. ... More
On weight complexes, pure functors, and detecting weightsDec 31 2018This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a nice description ... More
Fusion 2-categories and a state-sum invariant for 4-manifoldsDec 31 2018We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learningDec 31 2018Jan 29 2019The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive ... More
Distributions of Matching Distances in Topological Data AnalysisDec 29 2018In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques ... More
Vandermonde varieties, mirror spaces, and the cohomology of symmetric semi-algebraic setsDec 28 2018Let $\mathrm{R}$ be a real closed field, $d,k \in \mathbb{Z}_{> 0}$, $\mathbf{y} =(y_1,\ldots,y_d) \in \mathrm{R}^d$, and let $V_{d,\mathbf{y}}^{(k)}$ denote the Vandermonde variety defined by $p_1^{(k)} = y_1, \ldots, p_d^{(k)} = y_d$, where $p_j^{(k)} ... More
Neighboring mapping points theoremDec 28 2018Let f be a continuous map from a metric space X to M. We say that points in a subset N of X are f-neighbors if there exists a sphere S in M such that f(N) lies on S and there are no points of f(X) inside of S. We prove that if X is a unit sphere of any ... More
Tannaka duality for enhanced triangulated categories I: reconstructionDec 27 2018We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal{A}$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives ... More
Finite groups of rank two which do not involve $Qd(p)$Dec 27 2018Let $p>3$ be a prime. We show that if $G$ is a finite group with $p$-rank equal to 2, then $G$ involves $Qd(p)$ if and only if $G$ $p'$-involves $Qd(p)$. This allows us to use a version of Glauberman's ZJ-theorem to give a more direct construction of ... 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
Mod 2 cohomology ring of a kind of orbit configuration spaceDec 25 2018In this paper we caculate mod 2 cohomology ring of $F_{\mathbb{Z}_2^m}(\mathbb{R}^m,n)$ , which is local representation of orbit congfiguration spaces over small covers. We construct a differntial graded algebra, and there is a ring isomorphism between ... More
Nonconstant hexagon relations and their cohomologyDec 25 2018Jan 23 2019A construction of hexagon relations - algebraic realizations of four-dimensional Pachner moves - is proposed. It goes in terms of "permitted colorings" of 3-faces of pentachora (4-simplices), and its main feature is that the set of permitted colorings ... More
Möbius functions of directed restriction species and free operads, via the generalised Rota formulaDec 24 2018We present some tools for providing situations where the generalised Rota formula of arXiv:1801.07504 applies. As an example of this, we compute the M\"obius function of the incidence algebra of any directed restriction species, free operad, or more generally ... More
Analysis of contagion maps on a class of networks that are spatially embedded in a torusDec 24 2018A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion on a network whose nodes are ... More
Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic TopologyDec 23 2018While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In ... More
Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic TopologyDec 23 2018Feb 21 2019While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In ... More
Weight structures and the algebraic $K$-theory of stable $\infty$-categoriesDec 23 2018We introduce the notion of a bounded weight structure on a stable $\infty$-category and use this to prove the natural generalization of Waldhausen's sphere theorem: We show that the algebraic $K$-theory of a stable $\infty$-category with a bounded non-degenerate ... More
Pair component categories for directed spacesDec 22 2018The notion of a homotopy flow on a directed space was introduced in \cite{Raussen:07} as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy ... More
Persistence Bag-of-Words for Topological Data AnalysisDec 21 2018Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs are compact 2D representations formed by multisets of points. Their variable size makes them, however, ... More
Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limitDec 21 2018We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called "thermodynamic" regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized ... More
Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert spaceDec 20 2018We present a version of the equivariant gradient degree defined for equivariant gradient perturbations of an equivariant unbounded self-adjoint operator with purely discrete spectrum in Hilbert space. Two possible applications are discussed.
Homology groups of cubical sets with connectionsDec 18 2018Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of "connection" as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of ... More
Homomorphism Complexes and Maximal Chains in Graded PosetsDec 18 2018Dec 22 2018We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on ... More
Topological Data Analysis for the String LandscapeDec 17 2018Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string vacua. As a scaled-down ... More
Pseudo Maurer-Cartan perturbation algebra and pseudo perturbation lemmaDec 14 2018We introduce the pseudo Maurer-Cartan perturbation algebra, establish a structural result and explore the structure of this algebra. That structural result entails, as a consequence, what we refer to as the pseudo perturbation lemma. This lemma, in turn, ... More
Euclidean distance degree of the multiview varietyDec 13 2018The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry. It has direct applications in geometric modeling, computer vision, and statistics. We use non-proper Morse theory to give a topological interpretation ... More