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• Nikolaos Karaliolios # Nikolaos KaralioliosCEA · LIST

PhD

9
Publications
409
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39
Citations
Introduction
I do research AI, specializing in Computer Vision, in Pure Mathematics, specializing in Dynamical Systems, and in Numerical Analysis for Civil Engineering, specializing in Dynamics of Structures
November 2020 - present Position
• Engineer
Description
• I work on problems related to semi-supervised learning for computer vision
September 2019 - March 2020
Shift Technology
Position
• Researcher
Description
• I did research in AI and Machine Learning as part of Shift Technology's research team, specializing in computer vision applied in fraud detection and related problems in the insurance industry
September 2018 - August 2019
Université des Sciences et Technologies de Lille
Position
• PostDoc Position
Description
• Research on aspects of KAM theory and cohomology of dynamical systems

## Publications

Publications (9)
Article
Full-text available
In this m\'emoire we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty}$ reducible cocycles are dense in the $C^{\infty}$ topology, for a full measure set of frequencies. Moreover, we will show that every cocycle (or an...
Article
Full-text available
We continue our study of the local theory for quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $G=SU(2)$, over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the s...
Article
Full-text available
We provide a general argument for the failure of Anosov-Katok-like constructions to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A $C^{\infty }$ smooth diffeomorphism $f$ of a compact manifold $M$ is Cohomologically Rigid iff the equation, known as Linear Cohomological one, \begin{equation*} \psi \circ f - \psi = \va...
Article
Full-text available
Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d}\times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly redu...
Preprint
Full-text available
We prove that if the frequency of the quasi-periodic $\mathrm{SL}(2,\R)$ cocycle is Diophantine, then the following properties are dense in the subcritical regime: for any $\frac{1}{2}<\kappa<1$, the Lyapunov exponent is exactly $\kappa$-H\"older continuous; the extended eigenstates of the potential have optimal sub-linear growth; and the dual oper...
Preprint
Full-text available
Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d} \times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly red...
Article
Full-text available
We prove a theorem asserting that, given a Diophantine rotation $\alpha$ in a torus $\mathbb{T} ^{d} \equiv \mathbb{R} ^{d} / \mathbb{Z} ^{d}$, any perturbation, small enough in the $C^{\infty}$ topology, that does not destroy all orbits with rotation vector $\alpha$ is actually smoothly conjugate to the rigid rotation. The proof relies on a K.A.M...
Article
Full-text available
We prove local genericity of Distributional Unique Ergodicity (DUE) for certain classes of quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$, extending and/or refining some preceding results in the field. We then derive some consequences for one-frequency cocycles. We also confirm in this context (and, therefore, in a manifold of dimension $... Article Full-text available We study close-to-constants quasiperiodic cocycles in$\mathbb{T} ^{d} \times G$, where$d \in \mathbb{N} ^{*} $and$G\$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisf...
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