arXiv subjects

Gabriel Katz

Publications and source records attributed to Gabriel Katz.

At least 19 recordsLinked to original sources

arXiv2025

Contact de Rham cohomology and Hodge structures transversal to the Reeb foliations

Let $\beta$ be a contact form on a compact smooth manifold $X$ and $v_\beta$ its Reeb vector field. The paper applies general results of different authors about Hodge structures that are transversal to a given foliation to the special case of $1$-dimensional foliation generated by the Reeb flow $v_\beta$. The de Rham differential complex $\Omega_{\mathsf{basic}}^\ast(X, v_\beta)$ of, so called, {\sf basic} relative to $v_\beta$-flow differential forms is in the focus of this investigation. By definition, the basic forms vanish when being contracted with $v_\beta$, and so do their differentials. We prove that under the change $\beta \leadsto \beta_1 = \beta +df$, where a function $f:X \to \mathbf R$ such that $df(v_\beta) > -1$, the differential complexes $\Omega_{\mathsf {basic}}^\ast(X, v_{\beta_1})$ and $\Omega_{\mathsf{basic}}^\ast(X, v_\beta)$ are canonically isomorphic. We investigate when the $2$-form $d\beta$ and its powers deliver nontrivial elements in the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_\beta)$ of the differential complex $\Omega_{\mathsf{basic}}^\ast(X, v_\beta)$. Answers to these questions contrast sharply in the cases of a closed $X$ and a $X$ with boundary. On the other hand, building on work of Ra\'{z}ny \cite{Raz}, we show that on a closed manifold $X$, equipped with a transversal to the Reeb flow Hodge structure that satisfies the {\it Basic Hard Lefschetz Property}, the basic de Rham cohomology $H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X, v_\beta)$ are topological invariants of $X$.

math.DG↗

arXiv2023

Recovering contact forms from boundary data

Let $X$ be a compact smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\beta$ admit Lyapunov functions $f$. We tackle the question: how to recover $X$ and $\beta$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, \beta)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_\beta$-flow induced ``shadows" on the boundary $\partial X$. We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_\beta$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry.

math.SG↗

arXiv2023

On immersions and embeddings with trivial normal line bundles

Let $Z$ be a smooth compact $(n+1)$-manifold. We study smooth embeddings and immersions $\beta: M \to Z$ of compact or closed $n$-manifolds $M$ such that the normal line bundle $\nu^\beta$ is trivialized. For a fixed $Z$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this equivalence relation ``{\sf quasitopy}". It comes in two flavors: $\mathsf{IMM}(Z)$ and $\mathsf{EMB}(Z)$, based on immersions and embeddings into $Z$, respectively. We prove that the natural map $\mathsf{A}:\mathsf{EMB}(Z) \to \mathsf{IMM}(Z)$ is injective and admits a right inverse $\mathsf{R}:\mathsf{IMM}(Z) \to \mathsf{EMB}(Z)$, induced by the resolution of self-intersections. As a result, we get a map $$\mathcal B\Sigma:\; \mathsf{IMM}(Z) \big/ \mathsf{A}(\mathsf{EMB}(Z)) \longrightarrow \bigoplus_{k \in [2, n+1]} \mathbf B_{n+1-k}(Z)$$ whose target is a collection of smooth bordism groups of the space $Z$ and which differentiate between immersions and embeddings.

math.GT↗

arXiv2023

Doodles and blobs on a lined page: convex quasi-envelops of traversing flows on surfaces

Let $A$ denote the cylinder $\mathbb R \times S^1$ or the band $\mathbb R \times I$, where $I$ stands for the closed interval. We consider $2$-{\sf moderate immersions} of closed curves (``{\sf doodles}") and compact surfaces (``{\sf blobs}") in $A$, up to cobordisms that also are $2$-moderate immersions in $A \times [0, 1]$ of surfaces and solids. By definition, the $2$-moderate immersions of curves and surfaces do not have tangencies of order $\geq 3$ to the fibers of the obvious projections $A \to S^1$,\; $A \times [0, 1] \to S^1 \times [0, 1]$ or $A \to I$,\; $A \times [0, 1] \to I \times [0, 1]$. These bordisms come in different flavors: in particular, we consider one flavor based on {\sf regular embeddings} of doodles and blobs in $A$. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on $A= \mathbb R \times I$, our computations of $2$-moderate immersion bordisms $\mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)$ are near complete: we show that they can be described by an exact sequence of abelian groups $$0 \to \mathbf K \to \mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)\big/\mathbf{OC}^{\mathsf{emb}}_{\mathsf{moderate \leq 2}}(A) \stackrel{\mathcal I \rho}{\longrightarrow} \mathbb Z \times \mathbb Z \to 0,$$ where $\mathbf{OC}^{\mathsf{emb}}_{\mathsf{moderate \leq 2}}(A) \approx \mathbb Z \times \mathbb Z$, the epimorphism $\mathcal I \rho$ counts different types of crossings of immersed doodles, and the kernel $\mathbf K$ contains the group $(\mathbb Z)^\infty$ whose generators are described explicitly.

math.GT↗

arXiv2023

Algebras of smooth functions and holography of traversing flows

Let $X$ be a smooth compact manifold and $v$ a vector field on $X$ which admits a smooth function $f: X \to \mathbf R$ such that $df(v) > 0$. Let $\partial X$ be the boundary of $X$. We denote by $C^\infty(X)$ the algebra of smooth functions on $X$ and by $C^\infty(\partial X)$ the algebra of smooth functions on $\partial X$. With the help of $(v, f)$, we introduce two subalgebras $\mathcal A(v)$ and $\mathcal B(f)$ of $C^\infty(\partial X)$ and prove (under mild hypotheses) that $C^\infty(X) \approx \mathcal A(v) \hat\otimes \mathcal B(f)$, the topological tensor product. Thus the topological algebras $\mathcal A(v)$ and $\mathcal B(f)$, \emph{viewed as boundary data}, allow for a reconstruction of $C^\infty(X)$. As a result, $\mathcal A(v)$ and $\mathcal B(f)$ allow for the recovery of the smooth topological type of the bulk $X$.

math.GT↗

arXiv2022

Detecting intrinsic global geometry of an obstacle via layered scattering

Given a closed $k$-dimensional submanifold $K$, incapsulated in a compact domain $M \subset \mathbb E^n$, $k \leq n-2$, we consider the problem of determining the intrinsic geometry of the obstacle $K$ (like volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular $ε$-neighborhood $\mathsf T(K, ε)$ of $K$ in $M$. The geodesics that participate in this scattering emanate from the boundary $\partial M$ and terminate there after a few reflections from the boundary $\partial \mathsf T(K, ε)$. However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of $K$ by entering some trap there so that this ray will have infinitely many reflections from $\partial \mathsf T(K, ε)$. To rule out such a possibility, we modify the geometry of a tube $\mathsf T(K, ε)$ by building it from spherical bubbles. We need to use $\lceil \dim(K)/2\rceil$ many bubbling tubes $\{\mathsf T(K, ε_j)\}_j$ for detecting certain global invariants of $K$, invariants which reflect its intrinsic geometry. Thus the words "layered scattering" in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes $\mathsf T(K, ε)$ and their volumes.

math.DS↗

arXiv2022

Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

We study {\sf traversing} vector flows $v$ on smooth compact manifolds $X$ with boundary. For a given compact manifold $\hat X$, equipped with a traversing vector field $\hat v$ which is {\sf convex} with respect to $\partial\hat X$, we consider submersions/embeddings $α: X \to \hat X$ such that $\dim X = \dim \hat X$ and $α(\partial X)$ avoids a priory chosen tangency patterns $Θ$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hatγ$, we restrict the cardinality of $\hatγ\cap α(\partial X)$ by an even number $d$. We call $(\hat X, \hat v)$ a {\sf convex pseudo-envelop/envelop} of the pair $(X, v)$. Here the vector field $v = α^\dagger(\hat v)$ is the $α$-transfer of $\hat v$ to $X$. For a fixed $(\hat X, \hat v)$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $α: (X, v) \to (\hat X, \hat v)$, which we call a {\sf quasitopy}. The notion of quasitopy is a crossover between bordisms of pseudo-envelops and their pseudo-isotopies. In the study of quasitopies $\mathcal{QT}_d(Y, \mathbf cΘ)$, the spaces $\mathcal P_d^{\mathbf cΘ}$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $Θ$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat X, \hat v)$ and $\mathcal P_d^{\mathbf cΘ}$, the quasitopies of convex envelops which avoid the $Θ$-tangency patterns. We introduce characteristic classes of pseudo-envelops and show that they are invariants of their quasitopy classes. Then we prove that the quasitopies $\mathcal{QT}_d(Y, \mathbf cΘ)$ often stabilize, as $d \to \infty$.

math.GT↗

arXiv2022

Holography of geodesic flows, harmonizing metrics, and billiards' dynamics

Let $(M, g)$ be a Riemannian manifold with boundary, where $g$ is a non-trapping metric. Let $SM$ be the space of the spherical tangent to $M$ bundle, and $v^g$ the geodesic vector field on $SM$. We study the scattering maps $C_{v^g}: \partial^+_1SM \to \partial^-_1SM$, generated by the $v^g$-flow, and the dynamics of the billiard maps $B_{v^g, τ}: \partial^+_1SM \to \partial^+_1SM$, where $τ$ denotes an involution, mimicking the elastic reflection from the the boundary $\partial M$. We getting a variety of holography theorems that tackle the inverse scattering problems for $C_{v^g}$ and theorems that describe the dynamics of $B_{v^g, τ}$. Our main tools are a Lyapunov function $F: SM \to \mathbb R$ for $v^g$ and a special harmonizing Riemannian metrics $g^\bullet$ on $SM$, a metric in which $dF$ is harmonic. For such metrics $g^\bullet$, we get a family of isoperimetric inequalities of the type $vol_{g^\bullet}(SM) \leq vol_{g^\bullet |}(\partial(SM))$ and formulas for the average volume of the minimal hypesufaces $\{F^{-1}(c)\}_{c \in F(SM)}$. We investigate the interplay between the harmonizing metrics $g^\bullet$ and the classical Sasaki metric $gg$ on $SM$. Assuming ergodicity of $B_{v^g, τ}$, we also get Santaló-Chernov type formulas for the average length of free geodesic segments in $M$ and for the average variation of the Lyapunov function $F$ along the $v^g$-trajectories.

math.DS↗

arXiv2022

Detecting intrinsic global geometry of an obstacle via the layered scattering

Given a compact $k$-dimensional submanifold $K \subset \mathbf R^n$, incapsulated in a compact domain $M \subset \mathbf R^n$, we consider the problem of determining the inner geometry of the obstacle $K$ from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular $ε$-neighborhood $\mathsf T(K, ε)$ of $K$ in $M$. The geodesics emanate from $\partial M$ and terminate there, after a number of reflections from the boundary $\partial \mathsf T(K, ε)$. We use $\lceil \dim(K)/2\rceil$ many tubes $\{\mathsf T(K, ε_j)\}_j$ for detecting certain global intrinsic geometry invariants of $K$, thus the words "layered scattering" in the title. These invariants were studied by Hermann Weyl in his theory of tubes.

math.DS↗

arXiv2022

Spaces of polynomials as Grassmanians for immersions and embeddings

Let $Y$ be a smooth compact $n$-manifold. We study smooth embeddings and immersions $β: M \to \mathbb R \times Y$ of compact $n$-manifolds $M$ such that $β(M)$ avoids some a priory chosen closed poset $Θ$ of {\sf tangent patterns} to the fibers of the obvious projection $π: \mathbb R \times Y \to Y$. Then, for a fixed $Y$, we introduce an equivalence relation between such $β$'s; it is a crossover between pseudo-isotopies and bordisms. We call this relation {\sf quasitopy}. In the study of quasitopies, the spaces $\mathcal P_d^{\mathbf cΘ}$ of real univariate polynomials of degree $d$ with real divisors, whose combinatorial patterns avoid a given closed poset $Θ$, play the classical role of Grassmanians. We compute the quasitopy classes $\mathcal{QT}_d^{\mathsf{emb}}(Y, \mathbf cΘ)$ of $Θ$-constrained embeddings $β$ in terms of homotopy/homology theory of spaces $Y$ and $\mathcal P_d^{\mathbf cΘ}$. We prove also that the quasitopies of emeddings stabilize, as $d \to \infty$.

math.GT↗

arXiv2021

Spaces of polynomials with constrained real divisors, II. (Co)homology & stabilization

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree which have no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces P^{cΘ}_d of real monic univariate polynomials of degree d whose real divisors avoid given sequences of root multiplicities. These forbidden sequences are taken from an arbitrary poset Θof compositions that are closed under certain natural combinatorial operations. We reduce the computation of the homology H_*(P^{cΘ}_d) to the computation of the homology of a differential complex, defined purely combinatorially in terms of the given closed poset Θ. We also obtain the stabilization results about H^\ast(P^{c Θ}_d), as d goes to infinity. These results are deduced from our description of the homology of spaces B^{c Θ}_d whose points are binary real homogeneous forms, considered up to projective equivalence, with similarly Θ-constrained real divisors. In particular, we exhibit differential complexes that calculate the homology of these spaces and obtain some stabilization results for H^*(B^{c Θ}_d), as d goes to infinity. In particular, we compute the homology of the discriminants of projectivized binary real forms for which there is at least one line on which the form vanishes with multiplicity >= 2 and of their complements in \cB_d \cong RP^d.

math.AT↗

arXiv2021

Varieties in Cages: a Little Zoo of Algebraic Geometry

A $d^{\{n\}}$-cage $\mathsf K$ is the union of $n$ groups of hyperplanes in $\Bbb P^n$, each group containing $d$ members. The hyperplanes from the distinct groups are in general position, thus producing $d^n$ points, where hyperplanes from all groups intersect. These points are called the nodes of $\mathsf K$. We study the combinatorics of nodes that impose independent conditions on the varieties $X \subset \Bbb P^n$ containing them. We prove that if $X$, given by homogeneous polynomials of degrees $\leq d$, contains the points from such a special set $\mathsf A$ of nodes, then it contains all the nodes of $\mathsf K$. Such a variety $X$ is very special: in particular, $X$ is a complete intersection.

math.AG↗

arXiv2020

Applying Gromov's Amenable Localization to Geodesic Flows

Let $M$ be a compact smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincaré duality to study the {\sf traversally generic} geodesic flows on $SM$, the space of the spherical tangent bundle. Such flows generate stratifications of $SM$, governed by rich universal combinatorics. The stratification reflects the ways in which the flow trajectories are tangent to the boundary $\partial(SM)$. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension $k$ in terms of the normed homology $H_k(M; \mathbf R)$ and $H_k(DM; \mathbf R)$, where $DM = M\cup_{\partial M} M$ denotes the double of $M$. The norms here are the {\sf simplicial semi-norms} in homology. The more complex the metric on $M$ is, the more numerous the strata of $SM$ and $S(DM)$ are. %So one may regard our estimates as analogues of the Morse inequalities for the geodesics on manifolds with boundary. It turns out that the normed homology spaces form obstructions to the existence of globally $k$-{\sf convex} traversally generic metrics on $M$. We also prove that knowing the geodesic scattering map on $M$ makes it possible to reconstruct the stratified topological type of the space of geodesics, as well as the amenably localized Poincaré duality operators on $SM$.

math.GT↗

arXiv2020

Causal Holography of Traversing Flows

We study smooth {\sf traversing} vector fields $v$ on compact manifolds $X$ with boundary. A traversing $v$ admits a Lyapunov function $f: X \to \Bbb R$ such that $df(v) > 0$. We show that the trajectory spaces $\mathcal T(v)$ of {\sf traversally generic} $v$-flows are {\sf Whitney stratified spaces}, and thus admit triangulations amenable to their natural stratifications. Despite being spaces with singularities, $\mathcal T(v)$ retain some residual smooth structure of $X$. Let $\mathcal F(v)$ denote the oriented $1$-dimensional foliation on $X$, produced by a traversing $v$-flow. With the help of a {\sf boundary generic} $v$, we divide the boundary $\partial X$ of $X$ into two complementary compact manifolds, $\partial^+X(v)$ and $\partial^-X(v)$. Then, for a traversing $v$, we introduce the {\sf causality map} $C_v: \partial^+X(v) \to \partial^-X(v)$. Our main result claims that, for boundary generic traversing vector fields $v$, the causality map $C_v$ is allows for a reconstruction of the pair $(X, \mathcal F(v))$, up to a homeomorphism $Φ: X \to X$ such that $Φ|_{\partial X} = id_{\partial X}$. In other words, for a massive class of ODEs, we show that the topology of their solutions, satisfying a given boundary value problem, is {\sf rigid}. We call these results ``{\sf holographic}" since the $(n+1)$-dimensional $X$ and the un-parameterized dynamics of the $v$-flow are captured by a single map $C_v$ between two $n$-dimensional screens, $\partial^+X(v)$ and $\partial^-X(v)$. This holography of traversing flows has numerous applications to the dynamics of general flows. Some of them are described in the paper. Others, are just outlined.

math.GT↗

arXiv2019

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 \times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.

math.AT↗

arXiv2018

Causal Holography in Application to the Inverse Scattering Problems

For a given smooth compact manifold $M$, we introduce an open class $\mathcal G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow $v^g$ on the spherical tangent bundle $SM \to M$ admits a Lyapunov function (so the $v^g$-flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics. For every $g \in \mathcal G(M)$, the geodesic scattering along the boundary $\partial M$ can be expressed in terms of the \emph{scattering map} $C_{v^g}: \partial_1^+(SM) \to \partial_1^-(SM)$. It acts from a domain $\partial_1^+(SM)$ in the boundary $\partial(SM)$ to the complementary domain $\partial_1^-(SM)$, both domains being diffeomorphic. We prove that, for a \emph{boundary generic} metric $g \in \mathcal G(M)$ the map $C_{v^g}$ allows for a reconstruction of $SM$ and of the geodesic foliation $\mathcal F(v^g)$ on it, up to a homeomorphism (often a diffeomorphism). Also, for such $g$, the knowledge of the scattering map $C_{v^g}$ makes it possible to recover the homology of $M$, the Gromov simplicial semi-norm on it, and the fundamental group of $M$. Additionally, $C_{v^g}$ allows to reconstruct the naturally stratified topological type of the space of geodesics on $M$.

math.GT↗

arXiv2018

The Disk-Based Origami Theorem and a Glimpse of Holography for Traversing Flows

This paper describes a mechanism by which a traversally generic flow $v$ on a smooth connected manifold $X$ with boundary produces a compact $CW$-complex $\mathcal T(v)$, which is homotopy equivalent to $X$ and such that $X$ embeds in $\mathcal T(v)\times \mathbf R$. The $CW$-complex $\mathcal T(v)$ captures some residual information about the smooth structure on $X$ (such as the stable tangent bundle of $X$). Moreover, $\mathcal T(v)$ is obtained from a simplicial \emph{origami map} $O: D^n \to \mathcal T(v)$, whose source space is a disk $D^n \subset \partial X$ of dimension $n = \dim(X) -1$. The fibers of $O$ have the cardinality $(n+1)$ at most. The knowledge of the map $O$, together with the restriction to $D^n$ of a Lyapunov function $f:X \to \mathbf R$ for $v$, make it possible to reconstruct the topological type of the pair $(X, \mathcal F(v))$, were $\mathcal F(v)$ is the $1$-foliation, generated by $v$. This fact motivates the use of "holography" in the title.

math.GT↗

arXiv2018

On Holographic Structures, Traversing Flows, and Exotic Spheres

Any traversally generic vector flow on a compact manifold $X$ with boundary leaves some residual structure on its boundary $\d X$. A part of this structure is the flow-generated causality map $C_v$, which takes a region of $\d X$ to the complementary region. By the Holography Theorem from \cite{K4}, the map $C_v$ allows to reconstruct $X$ together with the unparametrized flow. The reconstruction is a manifestation of holographic description of the flow. In the paper, we introduce and study the holographic structures on a given closed manifold $Y$, which mimics $\d X$. We generalize the Holography Theorem so that is stated in terms of fillable holographic structures on $Y$. Such structures are intimately linked with traversally generic vector flows on manifolds $X$ whose boundary is $Y$. We conclude with few observations about the richness of holographic structures on smooth exotic spheres.

math.GT↗