arXiv research

Study shows non-wandering, partially hyperbolic systems are ergodic.

We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…

Mind wandering (MW) is a ubiquitous phenomenon which reflects a shift in attention from task-related to task-unrelated thoughts. There is a need for intelligent interfaces that can reorient attention when MW is detected due to its detrimental effects on performance and productivity. In this paper, we propose a deep lea…

Simulates patient pathways to detect delayed rare disease diagnoses.

Study partially hyperbolic dynamics on 3-manifolds with quasi-isometric center.

New method improves combinatorial optimization by overcoming inefficient sampling.

Paper characterizes early-stage dementia signatures from sensor data.

The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.

In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering spacelike geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that monodromy of Margulis Space Times are "Anosov representations in non semi-simple Lie …

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expandi…

New online few-shot learning model for context-aware recognition.

Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? T…

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initi…

Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale-Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set …

I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can blow up as we wander off to spatial infinity and/or as we decrease time to some s…

Geodesic flow mixing on convex projective manifolds proven.

We prove a theorem on structural stability of smooth attractor-repellor endomorphisms of compact manifolds, with singularities. By attractor-repellor, we mean that the non-wandering set of the dynamics $f$ is the disjoint union of a repulsive compact subset with a hyperbolic attractor on which $f$ acts bijectively. The…

The study shows conditions for thermostats to have no conjugate points.

The paper classifies 3D contact partially hyperbolic diffeomorphisms.

Explains the Borromean rings, icosahedron, and Poincaré homology sphere.

If X is a proper CAT(-1)-space and $Γ$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$Γ$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wanderin…

This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding…

Study of transitivity in partially hyperbolic maps with expanding linear part.

Towards a future where machine learning systems will integrate into every aspect of people's lives, researching methods to interpret such systems is necessary, instead of focusing exclusively on enhancing their performance. Enriching the trust between these systems and people will accelerate this integration process. M…

Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.

Constructs flows on quotients of Lie groups for Anosov subgroups.

Time and Sales of corn futures traded electronically on the CME Group Globex are studied. Theories of continuous prices turn upside down reality of intra-day trading. Prices and their increments are discrete and obey lattice probability distributions. A function for systematic evolution of futures trading volume is pro…

We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the…

Approximates measures on curved spaces using Dirac measures.

The goal of this thesis is to investigate the potential of predictive modelling for football injuries. This work was conducted in close collaboration with Tottenham Hotspurs FC (THFC), the PGA European tour and the participation of Wolverhampton Wanderers (WW). Three investigations were conducted: 1. Predicting the rec…

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …

Mean exit times concentrate near equators and minimal hypersurfaces in high dimensions.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer th…

The paper refines Mather-Thurston theorems for flat connections in manifolds.