## About

88

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Introduction

Gabriel Katz an affiliate researcher at the Department of Mathematics, Massachusetts Institute of Technology. Gabriel does research in Geometric Topology, Singularity Theory, Dynamical Systems, and Educational Technology. His current project is "Morse Theory & Gradient Flows on Manifolds with Boundary" and "Holography & Homology of Traversing Flows".

Additional affiliations

September 1979 - August 1983

## Publications

Publications (88)

Given a closed [Formula: see text]-dimensional submanifold [Formula: see text], encapsulated in a compact domain [Formula: see text], [Formula: see text], we consider the problem of determining the intrinsic geometry of the obstacle [Formula: see text] (such as volume, integral curvature) from the scattering data, produced by the reflections of geo...

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 $\ep...

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} : ∂^+ SM → ∂^− SM , generated by the (v^g)-flow, and the dynamics of the billiard maps B_{v^g, τ} : ∂^+SM → ∂^+SM , where τ denot...

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 $\alpha(\partial X)$...

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 introdu...

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 avoi...

A \(d^{\{n\}}\)-cage \(\mathsf K\) is the union of n groups of hyperplanes in \(\mathbb P^n\), 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...

A d {n}-cage K is the union of n groups of hyperplanes in 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 K. We study the combinatorics of nodes that impose independent conditions on th...

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(...

We study the non-vanishing gradient-like vector fields v on smooth compact manifolds X with boundary. We call such fields traversing. The traversally generic vector fields form an open and dense subset in the space of all traversing vector fields on X.
In Theorem 2.2, we show that the trajectory spaces T(v) of such traversally generic v-flows are...

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 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 reflect...

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 P cΘ d of real monic univariate polynomials of degree d whose real divisors avoid seque...

We study smooth traversing vector fields v on compact manifolds X with boundary. A traversing v admits a Lyapunov function f : X → R such that df (v) > 0. We show that the trajectory spaces T (v) of traversally generic v-flows are Whitney stratified spaces, and thus admit triangulations amenable to their natural stratifications. Despite being space...

For a traversing vector field v on a compact (n + 1)-manifold X with boundary , we use closed v-invariant differential n-forms Θ to define measures µΘ on the boundary ∂X, such that the v-flow generated causality map Cv : ∂ + X(v) → ∂ − X(v) preserves µΘ. In combination with a µΘ-preserving involution τ : ∂X → ∂X, which maps ∂ − X(v) to ∂ + X(v), th...

For a traversing vector field $v$ on a compact $(n+1)$-manifold $X$ with boundary, we use closed $v$-invariant differential $n$-forms $\Theta$ to define measures $\mu_\Theta$ on the boundary $\partial X$, such that the $v$-flow generated causality map $C_v: \partial^+X(v) \to \partial^-X(v)$ preserves $\mu_\Theta$. In combination with a $\mu_\Theta...

This paper describes a mechanism by which a traversally generic flow v on a smooth connected (n+1)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex T(v), which is homotopy equivalent to X and such that X embeds in T(v)×R. The CW-complex T(v) captures some residual information about the smooth structure on X (such as...

We propose an approach to decision support systems (DSS) that starts with the user first making their own unassisted decision αU and providing this as an input to the algorithm. Then, if the algorithm disagrees with the user’s initial decision, it iteratively works with the user to converge on a common decision or at least make the user reconsider...

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 t...

This monograph is an account of the author's investigations of gradient vector flows on compact manifolds with boundary. Many mathematical structures and constructions in the book fit comfortably in the framework of Morse Theory and, more generally, of the Singularity Theory of smooth maps.
The geometric and combinatorial structures, arising from...

In the late 80s, V.~Arnold and V.~Vassiliev initiated the study of the topology 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 sequence...

In the late 80s, V. Arnold and V. Vassiliev initiated the study of the topology 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 P cΘ d of real monic univariate polynomials of degree d whose real divisors avoid s...

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 T (v), which is homotopy equivalent to X and such that X embeds in T (v) × R. The CW-complex T (v) captures some residual information about the smooth structure on X (such as the stable tangent bundle o...

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...

Let $M$ be a compact connected smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the 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...

For a given smooth compact manifold $M$, we introduce a massive 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 $SM \to M$ is traversing. Moreover, for every $g \in \mathcal G(M)$, the geodesic scattering along the boundary...

We study the non-vanishing gradient-like vector fields v on smooth compact manifolds X with boundary. We call such fields traversing. The traversally generic vector fields form an open and dense subset in the space of all traversing vector fields on X. In Theorem 2.2, we show that the trajectory spaces T (v) of such traversally generic v-flows are...

This paper is about gradient-like vector fields and flows they generate on smooth compact surfaces with boundary. We use this particular 2-dimensional setting to present and explain our general results about non-vanishing gradient-like vector fields on n-dimensional manifolds with boundary. We take advantage of the relative simplicity of 2-dimensio...

In this paper, we present our general results about non-vanishing gradient-like vector flows on $n$-dimensional manifolds with boundary in the context of the flows on $2$-dimensional surfaces with boundary. We take advantage of the relative simplicity of $2$-dimensional worlds to explain and popularize our approach to the Morse theory on smooth man...

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 univers...

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 e...

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 e...

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 centra...

In this paper we continue to study the traversally generic vector fields $v$ on smooth compact manifolds $X$ with boundary. We show that the trajectory spaces $\mathcal T(v)$ of such $v$-flows are Whitney stratified spaces. Despite being spaces with singularities, $\mathcal T(v)$ retain some residual smooth structure of $X$. \smallskip
Let $ \math...

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 traje...

In this paper, we continue our study of the trajectory spaces of traversally
generic flows on smooth compact manifolds with boundary.
With the help of a traversally generic field v on X, we divide its boundary
d_1X into two complementary compact manifolds, d_1^+X(v) and d_1^-X(v), which
share a common boundary d_2X(v). Then we introduce the causali...

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...

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 definiti...

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary ∂X by compact submanifolds {\d_j^\pm X(v)}_j, where codim(\d_j^\pm X(v)) = j . Our main point is that this stratification reflects the stratified convexity/ concavity of the boundary ∂X with respe...

The Shape of Algebra in the Mirrors of Mathematics is a unique text aiming to explain some elements of modern mathematics and to show its flavor and unity. It is neither a standard textbook nor a tour of algebra for a casual reader. Rather, it is an attempt to share authors' mathematical experiences and philosophy with readers who have more than a...

The Shape of Algebra in the Mirrors of Mathematics is a unique text aiming to explain some elements of modern mathematics and to show its flavor and unity. It is neither a standard textbook nor a tour of algebra for a casual reader. Rather, it is an attempt to share authors' mathematical experiences and philosophy with readers who have more than a...

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 [Formula: see text] which takes its values in A(G). Here Ind G (v) denotes the equivariant index of the field v, [Formula: see text] the v-in...

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} trajecto...

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
$$ \lambda + \sum_{k = 1}^d {\left[\prod_{j = 1}^k L_{kj}\right]} $$where L
kj
’s are linear forms in x, y, and z and λ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are c...

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 ca...

Let M<sup>3</sup> be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$ , of a harmonic map $f : M^{3} \rightarrow S^1$ with Morse-type singularities delivers the Thurston norm $\chi_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3;\mathbb{Z})$ . In particular, for a map f with...

Copi, Huterer, Starkman, and Schwarz introduced multipole vectors in a tensor context and used them to demonstrate that the first-year Wilkinson microwave anisotropy probe (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 o...

Let Dd,k denote the discriminant variety of degree d polynomials in one variable with at least one of its roots being of multiplicity ≥ k. We prove that the tangent cones to Dd,k span Dd,k − 1 thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to Dd,k in Dd,k − 1 is directly linked to t...

For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are st...

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 a...

. 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 a theorem of E.Calabi [C],...

This paper is an extension of [9], where we have established a fixed-point formula for the formal t-deformations ind,(D, M) of the G-index ind(D, M) (G being a compact Lie group) of a basic differential operator D on a closed G-manifold M. The most important examples of D are the Signature, Dirac, and Euler-Todd operators. This deformation ind,(D,...

Davis, J.F. and G. Katz, Equivariant semicharacteristics and induction, Topology and its Applica- tions 51 (1993) 41-52. The paper defines a semicharacteristic bordism invariant for manifolds equipped with an action of a finite group with specified isotropy. Induction results and the Burnside ring are applied to give homological restrictions on som...

The class of concentrated periodic diffeomorphisms g: M → M is introduced. Adiffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small(with respect to the period of g and the dimension of M) arc on the circle. In many ways, the cyclic action generated by such a g behaveson the one hand as a circle action and...

In this paper we prove that in the space of all continuous mappings of a K-dimensional compact space X into complex linear space Cn the imbedding F: X → Cn with the property "any complex continuous function on F(X) can be uniformly approximated by complex polynomials on Cn" form a dense subset of type Gδ, provided that k ≤ 2/3n.

## Projects

Projects (2)

This projects collects different aspects of the theory of univariate polynomials