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Let $X$ be a compact connected 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 prove that any odd-dimensional $X$ admits such a contact form. 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., Comment: 47 pages, 3 figures

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., Comment: 8 pages, 2 figures

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., Comment: 42 pages, 12 figures

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 $X$.

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 $\epsilon$-neighborhood $\mathsf T(K, \epsilon)$ 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 $\partial \mathsf T(K, \epsilon)$. 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 $\partial \mathsf T(K, \epsilon)$. To rule out such a possibility, we modify the geometry of a tube $\mathsf T(K, \epsilon)$ by building it from spherical bubbles. We need to use $\lceil \dim(K)/2\rceil$ many bubbling tubes $\{\mathsf T(K, \epsilon_j)\}_j$ 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 $\mathsf T(K, \epsilon)$ and their volumes., Comment: arXiv admin note: text overlap with arXiv:2108.05478

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 $\alpha: X \to \hat X$ such that $\dim X = \dim \hat X$ and $\alpha(\partial X)$ avoids a priory chosen tangency patterns $\Theta$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hat\gamma$, we restrict the cardinality of $\hat\gamma \cap \alpha(\partial X)$ 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 $v = \alpha^\dagger(\hat v)$ is the $\alpha$-transfer of $\hat v$ to $X$. For a fixed $(\hat X, \hat v)$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $\alpha: (X, v) \to (\hat X, \hat v)$, 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 $\mathcal{QT}_d(Y, \mathbf c\Theta)$, the spaces $\mathcal P_d^{\mathbf c\Theta}$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $\Theta$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat X, \hat v)$ and $\mathcal P_d^{\mathbf c\Theta}$, the quasitopies of convex envelops which avoid the $\Theta$-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 $\mathcal{QT}_d(Y, \mathbf c\Theta)$ often stabilize, as $d \to \infty$.

Let $Y$ be a smooth compact $n$-manifold. We study smooth embeddings and immersions $\beta: M \to \mathbb R \times Y$ of compact $n$-manifolds $M$ such that $\beta(M)$ avoids some a priory chosen closed poset $\Theta$ of {\sf tangent patterns} to the fibers of the obvious projection $\pi: \mathbb R \times Y \to Y$. Then, for a fixed $Y$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this relation {\sf quasitopy}. In the study of quasitopies, the spaces $\mathcal P_d^{\mathbf c\Theta}$ of real univariate polynomials of degree $d$ with real divisors, whose combinatorial patterns avoid a given closed poset $\Theta$, play the classical role of Grassmanians. We compute the quasitopy classes $\mathcal{QT}_d^{\mathsf{emb}}(Y, \mathbf c\Theta)$ of $\Theta$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces $Y$ and $\mathcal P_d^{\mathbf c\Theta}$. We prove also that the quasitopies of emeddings stabilize, as $d \to \infty$., Comment: 49 pages, 4 figures

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^{c\Theta}_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 \Theta of compositions that are closed under certain natural combinatorial operations. We reduce the computation of the homology H_*(P^{c\Theta}_d) to the computation of the homology of a differential complex, defined purely combinatorially in terms of the given closed poset \Theta. We also obtain the stabilization results about H^\ast(P^{c \Theta}_d), as d goes to infinity. These results are deduced from our description of the homology of spaces B^{c \Theta}_d whose points are binary real homogeneous forms, considered up to projective equivalence, with similarly \Theta-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 \Theta}_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.

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 $\epsilon$-neighborhood $\mathsf T(K, \epsilon)$ of $K$ in $M$. The geodesics emanate from $\partial M$ and terminate there, after a number of reflections from the boundary $\partial \mathsf T(K, \epsilon)$. We use $\lceil \dim(K)/2\rceil$ many tubes $\{\mathsf T(K, \epsilon_j)\}_j$ 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., Comment: a mistake in the proof of Lemma 3.1

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 $B_{v^g, \tau}: \partial^+_1SM \to \partial^+_1SM$, where $\tau$ 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 $B_{v^g, \tau}$. 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 $B_{v^g, \tau}$, we also get Santal\'{o}-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., Comment: 48 pages, 3 figures

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., Comment: 17 pages, 5 figures

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., Comment: minor revision, to appear in Journal of Topology and Analysis

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., Comment: 12 pages, 2 figures. arXiv admin note: text overlap with arXiv:1409.0588

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., Comment: 21 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1409.0588

Let $M$ be a compact smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincar\'{e} 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 Poincar\'{e} duality operators on $SM$., Comment: 16 pages

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 $M$., Comment: 41 pages, 3 figures. arXiv admin note: text overlap with arXiv:1409.0588

In this paper, we present our general results about traversing flows on manifolds with boundary in the context of the flows on surfaces with boundary. We take advantage of the relative simplicity of $2D$-worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary, in which the boundary effects take the central stage., Comment: 40 pages, 16 figures

We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the trajectories that are tangent to the boundary in a particular canonical fashion. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension. These universal bounds are basically expressed in terms of the normed homology of the fundamental groups $\pi_1(D(X))$, where $D(X)$ denotes the double of $X$. The norm here is the Gromov simplicial semi-norm in homology. It turns out that some close relatives of the normed spaces $H_\ast(D(X); \R)$ form obstructions to the existence of $k$-convex traversally generic vector flows on $X$., Comment: 41 pages, 3 figures

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds., Comment: 16 pages, 5 figures

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 $\Phi: X \to X$ such that $\Phi|_{\partial X} = id_{\partial X}$. 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., Comment: 47 pages, 6 figures

This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic flows} on $(n+1)$-manifolds $X$, we embark on a detailed and somewhat tedious study of universal combinatorics of their tangency patterns with respect to the boundary $\d X$. This combinatorics is captured by a universal poset $\Omega^\bullet_{'\langle n]}$ which depends only on the dimension of $X$. It is intimately linked with the combinatorial patterns of real divisors of real polynomials in one variable of degrees which do not exceed $2(n+1)$. Such patterns are elements of another natural poset $\Omega_{\langle 2n+2]}$ that describes the ways in which the real roots merge, divide, appear, and disappear under deformations of real polynomials. The space of real degree $d$ polynomials $\mathcal P^d$ is stratified so that its pure strata are cells, labelled by the elements of the poset $\Omega_{\langle d]}$. This cellular structure in $\mathcal P^d$ is interesting on its own right (see Theorem \ref {th4.1} and Theorem \ref {th4.2}). Moreover, it helps to understand the \emph{localized} structure of the trajectory spaces $\mathcal T(v)$ for traversally generic fields $v$, the main subject of Theorem \ref {th5.2} and Theorem \ref {th5.3}., Comment: 37 pages, 6 figures

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and traversally generic vector fields $v$ are inspired by some classical notions from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that theory (cf. \cite{Morin}), we establish local versal algebraic models for the way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2\cdot \dim(X)$., Comment: 43 pages, 2 figures

As has been observed by Morse \cite{Mo}, any generic vector field $v$ on a compact smooth manifold $X$ with boundary gives rise to a stratification of the boundary $\d X$ by compact submanifolds $\{\d_j^\pm X(v)\}_{1 \leq j \leq \dim(X)}$, where $\textup{codim}(\d_j^\pm X(v))= j$. Our main observation is that this stratification reflects the stratified convexity/concavity of the boundary $\d X$ with respect to the $v$-flow. We study the behavior of this stratification under deformations of the vector field $v$. We also investigate the restrictions that the existence of a convex/concave traversing $v$-flow imposes on the topology of $X$. Let $v_1$ be the orthogonal projection of $v$ on the tangent bundle of $\d X$. We link the dynamics of the $v_1$-flow on the boundary with the property of $v$ in $X$ being convex/concave. This linkage is an instance of more general phenomenon that we call "holography of traversing fields"---a subject of a different paper to follow., Comment: 36 pages, 7 figures

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k \chi^G(\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equivariant index of the field v, {\d_k^+X\} the v-induced Morse stratification (see [M]) of the boundary \d X, and \chi^G(\d_k^+X) the class of the (n - k)-manifold \d_k^+X in $A(G)$. We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X \subset \R^n defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas., Comment: 13 pages

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in $x, y$ and $z$ and $\lambda$ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called \emph{multipole} vectors of $P$. In this paper we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theorems (which rely on our understanding of the Laplace operator $\Delta_{S^2}$) to more general differential operators $\Delta_Q$ that are constructed with the help of the quadratic form $Q(x, y, z)$. Then we introduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of $K(\pi, 1)$-spaces, where $\pi$ runs over a variety of modified braid groups., Comment: 43 pages, 3 figures. The new version of my paper contains references to the paper of V. Arnold Topological Content of the Maxwell Theorem on Multipole Representation of Spherical Functions which was overlooked in the previous version

We notice that a generic nonsingular gradient field $v = \nabla f$ on a compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$ of $X$. We study the transformations of $K(f, v)$ that are induced by deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$ with counting the \emph{double-tangent} trajectories of the $v$-flow, i.e. the trajectories that are tangent to the boundary $\d X$ at a pair of distinct points. Let $gc(X)$ be the minimum number of such trajectories, minimum being taken over all nonsingular $v$'s. We call $gc(X)$ the \emph{gradient complexity} of $X$. Next, we prove that there are only finitely many $X$ of bounded gradient complexity, provided that $X$ is irreducible and boundary irreducible with no essential annuli. In particular, there exists only finitely many hyperbolic manifolds $X$ with bounded $gc(X)$. For such $X$, their normalized hyperbolic volume gives an upper bound of $gc(X)$. If an irreducible and boundary irreducible $X$ with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then $X$ is a standard ball. All these and many other results of the paper rely on a careful study of the stratified geometry of $\d X$ relative to the $v$-flow. It is characterized by failure of $\d X$ to be \emph{convex} with respect to a generic flow $v$. It turns out, that convexity or its lack have profound influence on the topology of $X$. This interplay between intrinsic concavity of $\d X$ with respect to any gradient-like flow and complexity of $X$ is in the focus of the paper., Comment: 60 pages, 35 figures, pdfsync.sty, graphix.sty

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in $x, y$ and $z$ and $\lambda_k$'s are real numbers. The coefficients of these linear forms, viewed as 3D vectors, are called \emph{multipole} vectors of $P$. In this paper we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theorems (which rely on our understanding of the Laplace operator $\Delta_{S^2}$) to more general differential operators $\Delta_Q$ that are constructed with the help of the quadratic form $Q(x, y, z)$. Then we introduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of $K(\pi, 1)$-spaces, where $\pi$ runs over a variety of modified braid groups., Comment: 42 pages, 3 figures

The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families of cardinality d. The union of all lines from both families is called a cage, and the intersection X consists of d^2 points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. We study the subsets A of the nodal set X such that any plane algebraic curve C of degree d that contains A must contain X as well. As a corollary, we get a few generalizations of the famous Pascal theorem, generalizations that employ polygons (instead of hexagons as in Pascal's theorem) inscribed in a quadratic curve. Our results are closely related to the classical theorems of Chasles and Bacharach. Although the nodal sets X produced by our cages are quite special in comparison to more general complete intersections studied by Bacharach, the cages provide us with a much better grip on the combinatorics of subsets A of X with the property that is described above., Comment: 16 pages, 9 figures in the EPS format, and boxedeps.sty macro

Copi, Huterer, Starkman and Schwarz introduced multipole vectors in a tensor context and used them to demonstrate that the first-year WMAP quadrupole and octopole planes align at roughly the 99.9% confidence level. In the present article the language of polynomials provides a new and independent derivation of the multipole vector concept. Bezout's Theorem supports an elementary proof that the multipole vectors exist and are unique (up to rescaling). The constructive nature of the proof leads to a fast, practical algorithm for computing multipole vectors. We illustrate the algorithm by finding exact solutions for some simple toy examples, and numerical solutions for the first-year WMAP quadrupole and octopole. We then apply our algorithm to Monte Carlo skies to independently re-confirm the estimate that the WMAP quadrupole and octopole planes align at the 99.9% level., Comment: Version 1: 6 pages. Version 2: added uniqueness proof to Corollary 2; added proper citation (to Starkman et al.) for Open Question; other minor improvements

Let $\mathcal D_{d,k}$ denote the discriminant variety of degree $d$ polynomials in one variable with at least one of its roots being of multiplicity $\geq k$. We prove that the tangent cones to $\mathcal D_{d,k}$ span $\mathcal D_{d,k-1}$ thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to $\mathcal D_{d,k}$ in $\mathcal D_{d,k-1}$ is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation $P(z) = 0$ turns out to be equivalent to finding hyperplanes through a given point $P(z)\in \mathcal D_{d,1} \approx \A^d$ which are tangent to the discriminant hypersurface $\mathcal D_{d,2}$. We also connect the geometry of the Vi\`{e}te map $\mathcal V_d: \A^d_{root} \to \A^d_{coef}$, given by the elementary symmetric polynomials, with the tangents to the discriminant varieties $\{\mathcal D_{d,k}\}$. Various $d$-partitions $\{\mu\}$ provide a refinement $\{\mathcal D_\mu^\circ\}$ of the stratification of $\A^d_{coef}$ by the $\mathcal D_{d,k}$'s. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata $\{\mathcal D_\mu^\circ\}$., Comment: 43 pages, 12 figures

Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \emph{harmonic} map $f: M^3 \to S^1$ with Morse-type singularities delivers the Thurston norm $\chi_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\Sigma \subset M$ in the homology class of a fiber, we show that the Thurston number $\chi_-(\Sigma)$ satisfies an inequality $$\chi_-(\Sigma) \geq \chi_-(F_{best}) - \rho^\circ(\Sigma, f)\cdot Var_{\chi_-}(f).$$ Here the variation $Var_{\chi_-}(f)$ is can be expressed in terms of the $\chi_-$-invariants of the fiber components, and the twist $\rho^\circ(\Sigma, f)$ measures the complexity of the intersection of $\Sigma$ with a particular set $F_R$ of "bad" fiber components. This complexity is tightly linked with the optimal "$\tilde f$-height" of $\Sigma$, being lifted to the $f$-induced cyclic cover $\tilde M^3 \to M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its \emph{twist} $\rho_{\chi_-}(f)$. We prove that, for a harmonic $f$, $\chi_-([F_{best}]) = \chi_-(F_{best})$, if and only if, $\rho_{\chi_-}(f) = 0$., Comment: 13 figures

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use, in an essential way, a theorem of E.Calabi characterizing 1-forms which are harmonic with respect to some metric. We also study some interesting examples illustrating our results., Comment: 16 pages, AMSTex, 12 figures