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Homotopy Type of Independence Complexes of Certain Families of GraphsMay 16 2019We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes ... More
Defect of Euclidean distance degreeMay 16 2019Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the algebraic variety. ... More
The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrumMay 16 2019We specify exterior generators for $\pi_* THH(MU) = \pi_*(MU) \otimes E(\lambda'_n \mid n\ge1)$ and $\pi_* THH(BP) = \pi_*(BP) \otimes E(\lambda_n \mid n\ge1)$, and calculate the action of the $\sigma$-operator on these graded rings. In particular, $\sigma(\lambda'_n) ... More
2-Segal objects and algebras in spansMay 16 2019We define a category parameterizing Calabi-Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in ... More
On the profinite homotopy type of log schemesMay 15 2019We complete the program, initiated in [6], to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous ... More
On the Universal Property of Derived ManifoldsMay 15 2019It is well known that any model for derived manifolds must form a higher category. In this paper, we propose a universal property for this higher category, classifying it up to equivalence. Namely, the $\infty$-category $\mathbf{DMfd}$ of derived manifolds ... More
On formal groups and geometric quantizationMay 15 2019In the theory of geometric quantization, the (cobordism classes of) complex projective spaces, regarded as symplectic manifolds, are to the (cobordism classes of) complex projective spaces, regarded as almost complex manifolds, as elementary symmetric ... More
Vietoris-Rips Persistent HomologyMay 15 2019Persistence diagrams are useful displays that give a summary information regarding the topological features of some phenomenon. Usually, only one persistence diagram is available, and replicated persistence diagrams are needed for statistical inference. ... More
On O'hara knot energies I: Regularity for critical knotsMay 15 2019We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the M\"obius energy. For the M\"obius energy, due to the celebrated work of Freedman, He, and Wang, ... More
Geometry of universal embedding spaces for almost complex manifoldsMay 15 2019We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was ... More
Hyperbolic Nodal Band Structures and Knot InvariantsMay 14 2019We extend the list of known band structure topologies to include hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the ... More
Stable components and layersMay 14 2019Component graphs $\Gamma_{0}(F)$ are defined for arrays of sets $F$. The path components of $\Gamma_{0}(F)$ are the stable components of the array $F$. The stable components for the system of Lesnick complexes $\{ L_{s,k}(X) \}$ for a finite data set ... More
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
Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-MultiplicitiesMay 14 2019We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a ... More
Stability of Loday constructionsMay 14 2019We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X \otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $\Sigma X$. We prove several structural properties ... More
Stability conditions on morphisms in a categoryMay 14 2019Let $\mathbf D$ be the homotopy category of a stable infinity category. Then the category $\mathbf D^{\Delta^1}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathbf D^{\Delta^1}}$ of stability conditions on $\mathbf D^{\Delta^1}$ is well-defined ... More
Unstable $ν_1$-Periodic Homotopy of Simply Connected, Finite $H$-Spaces, using Goodwillie CalculusMay 13 2019In this paper we recover Bousfield's computation of $\nu_1$-periodic homotopy groups of simply connected, finite $H$-spaces from \cite{Bou99} using the techniques of Goodwillie calculus. This is done through first computing Andr\'{e}-Quillen cohomology ... More
On the homology of the commutator subgroup of the pure braid groupMay 13 2019We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological ... More
Torus actions of complexity one in non-general positionMay 12 2019Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly ... More
Cohomology of configuration spaces of surfacesMay 12 2019We compute the rational cohomology of unordered configuration spaces of points on any closed orientable surface. We find a series with coefficients in the Grothendieck ring of the symplectic group sp(2g) that describes explicitly the decomposition of ... More
Real moduli space of stable rational curves revistedMay 11 2019We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use ... 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
Persistent homology of the sum metricMay 10 2019Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$ equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a module over a polynomial ... More
Motivic Mahowald invariants over general base fieldsMay 10 2019The motivic Mahowald invariant was introduced in \cite{Qui19a} and \cite{Qui19b} to study periodicity in the $\mathbb{C}$- and $\mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of characteristic ... More
$N_\infty$-operads and associahedraMay 09 2019We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the collection ... More
Classical shadows of stated skein representations at roots of unityMay 09 2019We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically ... More
Conformal nets V: dualizabilityMay 09 2019We prove that finite-index conformal nets are fully dualizable objects in the 3-category of conformal nets. Therefore, assuming the cobordism hypothesis applies, there exists a local framed topological field theory whose value on the point is any finite-index ... More
Higher Segal spaces and Lax $\mathbb{A}_\infty$-algebrasMay 08 2019The notion of a higher Segal space was introduced by Dyckerhoff and Kapranov as a general framework for studying higher associativity inherent in a wide range of mathematical objects. In the present work we formalize the connection between this notion ... More
$\overline{Spec\mathbb Z}$ and the Gromov normMay 08 2019We define the homology of a simplicial set with coefficients in a Segal's $\Gamma$-set ($\mathbf S$-module). We show the relevance of this new homology with values in $\mathbf S$-modules by proving that taking as coefficients the $\mathbf S$-modules at ... More
Multiplicativity and nonrealizable equivariant chain complexesMay 08 2019Let $G$ be a finite $p$-group and $\mathbb{F}$ a field of characteristic $p$. We filter the cochain complex of a free $G$-space with coefficients in $\mathbb{F}$ by powers of the augmentation ideal of $\mathbb{F} G$. We show that the cup product induces ... More
BigerbesMay 08 2019Bigerbes give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; ... More
Smooth classifying spaces for differential $K$-theoryMay 08 2019We construct a version of differential $K$-theory based on smooth Banach manifold models for the homotopy types $B \mathrm U\times Z$ and $\mathrm U$ that appear in the topological $K$-theory spectrum. These manifolds carry natural differential forms ... More
Universal spaces of parameters for complex Grassmann manifolds $G_{q+1,2}$May 08 2019Buchstaber and Terzic introduced a notion of universal space of parameters $\mathcal{F}$ for a manifold $M^{2n}$, which has an effective action of compact torus $T^k$ , $k \leq n$ with some additional properties. with special properties. This space is ... More
Bialgebraic approach to rack cohomologyMay 07 2019We interpret the complexes defining rack cohomology in terms of a certain differential graded bialgebra. This yields elementary algebraic proofs of old and new structural results for this cohomology theory. For instance, we exhibit two explicit homotopies ... More
Tautological Rings of FibrationsMay 07 2019We study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincar\'e fibre, i.e. the ring obtained by fibre integrating powers of the fibrewise Euler class. We approach this ring of characteristic classes using rational ... More
On a Conjecture of Mahowald on the Cohomology of Finite Sub-Hopf algebras of the Steenrod AlgebraMay 07 2019Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of ... More
On the reorderability of node-filtered order complexesMay 07 2019Growing graphs describe a multitude of developing processes from maturing brains to expanding vocabularies to burgeoning public transit systems. Each of these growing processes likely adheres to proliferation rules that establish an effective order of ... More
Orbit spaces of torus actions on Hessenberg varietiesMay 06 2019We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. ... More
String$\mathbf{^c}$ Structures and Modular InvariantsMay 06 2019In this paper, we study some algebraic topology aspects of String$^c$ structures, more precisely, from the aspect of Whitehead tower and the aspect of the loop group of $Spin^c(n)$. We also extend the generalized Witten genus constructed for the first ... More
Topological manifold bundles and the $A$-theory assembly mapMay 06 2019We give a new proof of an index theorem for fiber bundles of compact $topological$ manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through the assembly ... More
Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operatorMay 05 2019We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected ... More
Higher analogs of simplicial and combinatorial complexityMay 04 2019We introduce higher simplicial complexity of a simplicial complex $K$ and higher combinatorial complexity of a finite space $P$ (i.e. $P$ is a finite poset). We relate higher simplicial complexity with higher topological complexity of $|K|$ and higher ... More
On Goussarov-Polyak-Viro Conjecture of knots with degree threeMay 04 2019A knot invariant ordered by filtered finite dimensional vector spaces is called finite type. It has been conjectured that every finite type invariant of classical knots could be extended to a finite type invariant of long virtual knots (Goussarov-Polyak-Viro ... More
A dg Lie model for relative homotopy automorphismsMay 03 2019We construct a dg Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace.
The cohomology of the Steenrod algebra and the mod $p$ Lannes-Zarati homomorphismMay 02 2019In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $\varphi_3^{\mathbb{F}_p}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd ... More
New construction of the brane coproduct and vanishing of cup products on sphere spacesMay 02 2019Using the loop coproduct, Menichi proved that the cup product with the orientation class vanishes for a closed connected oriented manifold with non-trivial Euler characteristic. We generalize this to the sphere spaces, i.e. the mapping spaces from spheres, ... More
An algorithmic search for $\mathcal{A}$-annihilated classes in the Dyer-Lashof algebra and $H_*QS^0$ I. Closed form for low lengths and tables in low dimensionsMay 02 2019The aim of this work is to publicise some computational results involving tables which contain $\mathcal{A}$-annihilated monomials, excluding square classes, in the Dyer-Lashof algebra and $H_*QS^0$; our computations go up to dimension $1.1\times 10^7$ ... More
Coordinatizing Data With Lens Spaces and Persistent CohomologyMay 01 2019We introduce here a framework to construct coordinates in \emph{finite} Lens spaces for data with nontrivial 1-dimensional $\mathbb{Z}_q$ persistent cohomology, $q\geq 3$. Said coordinates are defined on an open neighborhood of the data, yet constructed ... More
Flag Bott manifolds of general Lie type and their equivariant cohomology ringsMay 01 2019In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant ... More
Exotic Multiplications on Periodic Complex BordismApr 30 2019Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of $BU$. This presents an $\mathbb{E}_\infty$ structure on periodic complex bordism by different means than the usual Thom spectrum ... More
$C_2$-equivariant Homology Operations: Results and FormulasApr 30 2019In this note we state corrected and expanded versions of our previous results on power operations for $C_2$-equivariant Bredon homology with coefficients in the constant Mackey functor on $\mathbb{F}_2$. In particular, we give a version of the Adem relations. ... More
Mod 2 power operations revisitedApr 30 2019In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $\mathbb{E}_{\infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing ... More
Toric topology of the Grassmannian of planes in $\mathbb{C}^5$ and the del Pezzo surface of degree fiveApr 30 2019We are complementing the work of Buchstaber and Terzi\'c by providing an alternative approach to determine the homology of the orbit space of a maximal compact torus action on the complex Grassmannian Gr(2,5). Our approach uses the well-known Geometric ... More
Toric topology of the Grassmannian of planes in $\mathbb{C}^5$ and the del Pezzo surface of degree $5$Apr 30 2019May 05 2019We are complementing the work of Buchstaber and Terzi\'c by providing an alternative approach to determine the homology of the orbit space of a maximal compact torus action on the complex Grassmannian Gr(2,5). Our approach uses the well-known Geometric ... More
Noncommutative Geometry of Quantized CoveringsApr 30 2019This research is devoted to the noncommutative generalization of topological coverings. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here ... More
Goodwillie calculus in the category of algebras over a chain complex operadApr 30 2019In this paper, we study the homotopy category $\text{Alg}_\mathcal{O}$ of algebras over a fixed and reduced operad $\mathcal{O}$ of non negatively graded chain complexes over a field of characteristic 0. We give an explicit description of homotopy pullbacks ... More
Real motivic and $C_2$-equivariant Mahowald invariantsApr 30 2019The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. In \cite{Qui17}, we studied an analog of this construction in the setting of motivic stable homotopy theory ... More
The positive scalar curvature cobordism categoryApr 29 2019We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, ... More
On morphisms killing weights and Hurewicz-type theoremsApr 29 2019We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most ... More
On stratification for spaces with Noetherian mod $p$ cohomologyApr 29 2019Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;\mathbb{F}_p)$ be the commutative ring spectrum of $\mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module ... 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
A Motion Planning Algorithm in a lollipop graphApr 29 2019May 01 2019This paper is concerned with problems relevant to motion planning in robotics. Configuration spaces are of practical relevance in designing safe control schemes for robots moving on a track. The topological complexity of a configuration space is an integer ... More
A Motion Planning Algorithm in a lollipop graphApr 29 2019This paper is concerned with problems relevant to motion planning in robotics. Configuration spaces are of practical relevance in designing safe control schemes for robots moving on a track. The topological complexity of a configuration space is an integer ... More
On certain complex surface singularitiesApr 29 2019The thesis deals with holomorphic germs $ \Phi: (\mathbb{C}^2, 0) \to (\mathbb{C}^3,0) $ singular only at the origin, with a special emphasis on the distinguished class of finitely determined germs. The results are published in two articles (arXiv:1404.2853 ... More
An estimate of the Hopf degree of fractional Sobolev mappingsApr 29 2019We estimate the Hopf degree for smooth maps $f$ from $\mathbb{S}^{4n-1}$ to $\mathbb{S}^{2n}$ in the fractional Sobolev space. Namely we show that for $s \in [1 - \frac{1}{4n}, 1]$ \[ \left |{\rm deg}_H(f)\right | \lesssim [f]_{W^{s,\frac{4n-1}{s}}}^{\frac{4n}{s}}. ... More
The Milnor-Moore theorem for $L_\infty$ algebras in rational homotopy theoryApr 29 2019We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing ... More
Controlled surgery and $\mathbb{L}$-homologyApr 29 2019This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $\sigma^c (f, ... More
On gauge groups over high dimensional manifolds and self-equivalences of $H$-spacesApr 28 2019Let $Y$ be a pointed space and let $\mathcal E(Y^r)$ be the group of based self-equivalences of $Y^r$, $r\geq 2$. For $Y$ a homotopy commutative $H$-group we construct a subgroup $\mathcal E_{\mathrm{Mat}}(Y^r)$ of $\mathcal E(Y^r)$ which has a group ... More
On the topology of bi-cyclopermutohedraApr 27 2019Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $\{1,\cdots, n+1\}$ up to cyclic permutations and orientation reversion. This poset is the ... More
Computing A-Homotopy Groups Using Coverings and Lifting PropertiesApr 26 2019In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes ... More
Configurations of noncollinear points in the projective planeApr 25 2019Apr 29 2019We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$ are computed ... More
Configurations of noncollinear points in the projective planeApr 25 2019We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$ are computed ... More
Shifted Coisotropic CorrespondencesApr 25 2019We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected ... More
On the automorphism group of the Morse complexApr 24 2019Let $K$ be a finite, connected, abstract simplicial complex. The Morse complex of $K$, first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient vector fields on $K$. We show that if $K$ is neither the boundary of the ... More
Coalgebraic Formal Curve Spectra and the Annular TowerApr 24 2019We import into homotopy theory the algebro-geometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava $K$-theory of height $d$, we show that this can be used to produce a choice-free ... More
Milnor invariants of braids and welded braids up to homotopyApr 24 2019We consider the group of pure welded braids (also known as loop braids) up to (link-)homotopy. The pure welded braid group classically identifies, via the Artin action, with the group of basis-conjugating automorphisms of the free group, also known as ... More
Sparse Nerves in PracticeApr 23 2019Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread ... More
Twisted Cohomotopy implies M-Theory anomaly cancellationApr 23 2019We show that all the expected anomaly cancellations in M-theory follow from charge-quantizing the C-field in the non-abelian cohomology theory twisted Cohomotopy. Specifically, we show that such cocycles exhibit all of the following: (1) the half-integral ... More
Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $\operatorname{S}$-constructionsApr 23 2019These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type $\mathbb{A}$ to Eilenberg--Mac ... More
$A_\infty$-Minimal Model on Differential Graded AlgebrasApr 23 2019For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. As an application, the $A_\infty$-algebras constructed by Tsai, Tseng ... More
The winding invariantApr 22 2019Every element $w$ in the commutator subgroup of the free group $\mathbb{F}_2$ of rank 2 determines a closed curve in the grid $\mathbb{Z} \times \mathbb{R} \cup \mathbb{R} \times \mathbb{Z} \subseteq \mathbb{R}^2$. The winding numbers of this curve around ... More
Pyknotic objects, I. Basic notionsApr 22 2019Pyknotic objects in are sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of a ... More
Pyknotic objects, I. Basic notionsApr 22 2019Apr 30 2019Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of ... More
Relative singular value decomposition and applications to LS-categoryApr 22 2019Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices. For any $1\leq k\leq n$, an element $A$ of $Sp(n)$ can be decomposed in $A= \begin{bmatrix} \alpha&T\cr \beta&P \end{bmatrix}$ with $P$ a $(k\times k)$-matrix. In this work, starting ... More
Monodromy in weight graphs and its applications to torus actionsApr 21 2019In this paper, torus actions on some particular non-singular hypersurfaces in non-singular projective toric varieties are studied. The generalised Buchstaber-Ray varieties $BR_{i,j}\subset BF_{i}\times \mathbb{P}^{j}$, $i,j\geq 0$, are introduced. The ... More
A General Neural Network Architecture for Persistence Diagrams and Graph ClassificationApr 20 2019Graph classification is a difficult problem that has drawn a lot of attention from the machine learning community over the past few years. This is mainly due to the fact that, contrarily to Euclidean vectors, the inherent complexity of graph structures ... More
Shuffle algebras and perverse sheavesApr 19 2019We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct ... More
Isotropic motivesApr 19 2019In this article we introduce the local versions of the Voevodsky category of motives with Z/p-coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservative ... More
Helly meets Garside and ArtinApr 19 2019A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, ... More
Rationally elliptic toric varietiesApr 18 2019We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
Rationally elliptic toric varietiesApr 18 2019Apr 25 2019We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described.
Abstract Goerss-Hopkins theoryApr 18 2019We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-category of synthetic spectra, this yields obstructions to realizing a comodule ... More
Abstract Goerss-Hopkins theoryApr 18 2019Apr 19 2019We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-category of synthetic spectra, this yields obstructions to realizing a comodule ... More
Abstract Goerss-Hopkins theoryApr 18 2019Apr 24 2019We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-category of synthetic spectra, this yields obstructions to realizing a comodule ... More
Aspherical completions and rationally inert elementsApr 18 2019Let $X$ be a connected space. An element $[f]\in \pi_n(X)$ is called rationally inert if $\pi_*(X)\otimes \mathbb Q \to \pi_*(X\cup_fD^{n+1})\otimes \mathbb Q$ is surjective. We extend the results obtained in the simply connected case, and prove in particular ... More
Sullivan completionsApr 18 2019The Sullivan construction associates to each path connected space or connected simplicial set, $X$, a special cdga, its minimal model $(\land V,d)$, and to each such cdga $\land W$ its geometric realisation $\langle \land W\rangle$. The composite of these ... More
Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutionsApr 17 2019In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for Erdoes-Renyi random ... More
Tangent of K-theoryApr 17 2019We show that the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. More precisely, we prove that the tangent of K-theory, in terms of (abelian) deformation problems over k, is cyclic ... More
On the neighborhood complex of $\vec{s}$-stable Kneser graphsApr 17 2019In 2002, A. Bj\"orner and M. de Longueville showed the neighborhood complex of the $2$-stable Kneser graph ${KG(n, k)}_{2-\textit{stab}}$ has the same homotopy type as the $(n-2k)$-sphere. A short time ago, an analogous result about the homotopy type ... More
Free commuting involutions on closed two-dimensional surfacesApr 17 2019We consider the function $f(g)$ that assigns to an orientable surface $M$ of genus $g$ the maximal number of free commuting independent involutions on $M$. We show that the surface of minimal genus $g$ with $f(g)=n$ is a real moment-angle complex $R_K$, ... More