# The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

@article{Katz2014TheSS, title={The Stratified Spaces of Real Polynomials \& Trajectory Spaces of Traversing Flows}, author={Gabriel Katz}, journal={arXiv: Geometric Topology}, year={2014} }

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…

## 17 Citations

### Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume

- Mathematics
- 2015

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…

### Applying Gromov's Amenable Localization to Geodesic Flows

- Mathematics
- 2017

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…

### Real polynomials with constrained real divisors. I. Fundamental groups

- Mathematics
- 2019

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…

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

- Mathematics
- 2021

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…

### REAL POLYNOMIALS WITH CONSTRAINED REAL DIVISORS, II. (CO)HOMOLOGY AND STABILIZATION

- Mathematics
- 2019

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 and with no real roots of multiplicity exceeding a given…

### Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

- Mathematics
- 2022

We study traversing vector flows v on smooth compact manifolds X with boundary. For a given compact manifold X̂, equipped with a traversing vector field v̂ which is convex with respect to ∂X̂, we…

### On Holographic Structures, Traversing Flows, and Exotic Spheres

- Mathematics
- 2018

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 $C_v$,…

### Causal Holography of Traversing Flows

- Mathematics
- 2014

We study smooth traversing vector fields v on compact manifolds X with boundary. A traversing v admits a Lyapunov function $$f: X \rightarrow \mathbb R$$ f : X → R such that $$df(v) > 0$$ d f ( v ) >…

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

- MathematicsQualitative Theory of Dynamical Systems
- 2020

This paper describes a mechanism by which a traversally generic flow v on a smooth connected $$(n+1)$$ ( n + 1 ) -dimensional manifold X with boundary produces a compact n -dimensional CW -complex…

### Causal holography in application to the inverse scattering problems

- MathematicsInverse Problems & Imaging
- 2019

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…

## References

SHOWING 1-2 OF 2 REFERENCES

### Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

- Mathematics
- 2014

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 by compact submanifolds , where . Our main…

### Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary

- Mathematics
- 2014

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…