A Bill of Rights is what the people are entitled to against every government, and what no just government should refuse, or rest on inference.
by Thomas Jefferson

Stratified affine spaces

These are quick notes for a joint work with Ch. Wacheux

Here, by an affine manifold, we mean a paracompact manifold with a locally flat affine structure. If the manifold is of dimension n, then near each point we have a chart whose coordinate functions are affine functions, and any local affine function is . . . → Read More: Stratified affine spaces

Notes on random systems 3: Liouville theorem for RDS

Extension of Liouville’s theorem to the case of RDS

The case without first integrals:

X_1, …, X_p are p generators of RDS on a compact manifold M of dimension p, which commute with each other, and whose deterministic parts are linearly independent everwhere. Then M is a  p-dimensional torus T^p with a periodic coordinate . . . → Read More: Notes on random systems 3: Liouville theorem for RDS

Notes on random systems 2

What is a random fixed point ?

A reference: Ochs – Oseledets: Examples of  RDS on a closed unit ball without random fixed points. (So the topological fixed point theorem is NOT valid for RDS)

Definition. A random fixed point of a RDS \Phi over noise space (\Omega, \theta) (theta is the dynamics in . . . → Read More: Notes on random systems 2

Notes on random systems 1: RDS vs NDS

Edited: 25/02/2014

These series of notes are for a research project that I’m doing with a student of mine. Some of what I write here will look quite stupid, because I myself know little about random dynamical systems.

I’ll explain why we’re interested in these systems later. But first, here . . . → Read More: Notes on random systems 1: RDS vs NDS

Monodromy can kill global convexity!

We just got a rather surprising result in our joint research project with Christophe Wacheux (currently post-doc at EPFL) about the intrinstic convexity of singular affine spaces.

The problem is to study the intrinsic local and global convexity of the affine structure of the base space of integrable Hamiltonian systems whose singularities are nondegenerate . . . → Read More: Monodromy can kill global convexity!

Thuật toán của người mất trí nhớ

Có một bài toán “khá đơn giản” sau về vấn đề thuật toán, tôi biết chắc chắn là giải được (vì có định lý về vấn đề này), có điều tôi thử tự tìm lời giải mà loay hoay mãi chưa ra:

Có một người ở một làng bị mất trí và được cho . . . → Read More: Thuật toán của người mất trí nhớ

Integrable p-vector fields and singular foliations

There seems to be a lot of confusion (among my colleagues, and also of myself) concerning the relationships between singular foliations and integrable p-vector fields (a.k.a. Nambu structures). The aim of this note is to make some clarifications.

1) How to construct a singular foliation from an integrable p-vector field ?

The obvious (but . . . → Read More: Integrable p-vector fields and singular foliations

Talk at AlanFest 07/2013

 

This week I’m at the AlanFest in EPFL, Switzerland, on the occasion of Alan Weinstein’s 70th birthday. I gave  a talk on Thursday entitled:

A normalization toolbox, with applications to singular foliations

Here are the slides of my talks (with some typographical errors), for people who might be interested:

NTZ_AlanFest2013

. . . → Read More: Talk at AlanFest 07/2013

Linearization of smooth integrable systems

I’m writing down here the ideas for proving that smooth nondegenerate integrable dynamical systems are smoothly linearizable. The analytic case can be proved using analytic torus actions (my paper about that will appear in Ergodic Th Dyn Sys). I think the smooth case is also true, but the proof is much more complicated than . . . → Read More: Linearization of smooth integrable systems

Anti-canonical bundle of singular foliations

Take a singular foliation say of dimension k.

Locally at each point there exists $k$ vector fields X1, …, Xk which are tangent to the foliation and which are linearly independent almost everywhere (we will only consider foliations which satisfy this property)

The wedge product L = X1…Xk is a Nambu structure tangent to . . . → Read More: Anti-canonical bundle of singular foliations

Linearization and stability of singular foliations

This is the topic that I want to talk about in the conference in honor of Alan Weinstein’s 70th birthday in EPFL (Lausanne) in July. This post is the place in keep the preparation for my talk.

A work of mine on the linearization of proper Lie groupoids was directly influenced by Alan (it . . . → Read More: Linearization and stability of singular foliations

Oracle phổ quát bậc 2 ?

Ok, đây là tôi tự bổ túc văn hóa về complexity thôi.

Trong lý thuyết complexity, người ta nói đến các oracle, tức là các “hộp đen” để tính các hàm (hay trả lời các câu hỏi) nào đó,  chỉ cần cho input (thuộc loại nào đó) vào đấy thì sẽ ra ngay output, . . . → Read More: Oracle phổ quát bậc 2 ?

Introduction to dynamical systems (1)

These are the notes for an optional course on dynamical systems that I will give to 4th year mathematics students this semester.

The course is in French, but for convenience I will write the notes in English. (In order to become competitive, my French students will have to learn to use English anyway).

The . . . → Read More: Introduction to dynamical systems (1)