If I were asked to enumerate ten educational stupidities, the giving of grades would head the list... If I can't give a child a better reason for studying than a grade on a report card, I ought to lock my desk and go home and stay there.
by Dorothy De Zouche

Book chapter on the geometry of integrable non-Hamiltonian systems

Just finished my chapter for a book on integrable systems. In case anyone is interested, here is the PDF file:


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

Three lectures on singular foliations

For those who might be interested: This PDF file contains the slides of my 3-hour minicourse on singular foliations, given in the Workshop “GAP XII: Geometric Mechanics”, Internatinal Mathematical Forum of Tsing Hua University, Sanya, 10-14/March/2014:


The main topic of this minicourse is the correspondence Singular Foliations <–> Nambu structures, and how . . . → Read More: Three lectures on singular foliations

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

Commuting Foliations

Our paper with Truong Hong Minh entitled “Commuting Foliations” has been accepted for publication in a special isue dedicated to Alain Chenciner in the journal  Regular and Chaotic Dynamics

The most imporant part of this paper is actually the exposition of the relationship between singular foliations and Nambu structures: how to go from a . . . → Read More: Commuting Foliations

Geometry and Physics XII, Hainan (China), 10-14/03/2014

Next year I’ll come to Sanya, Hainan, China to attend “GAP XII”, an international conference on geometry and physics.

The organizers said they would have money to cover local expenses for a number of participants, so if anyone is interested in visiting Sanya, please contact them (or contact me if you don’t know them)

. . . → Read More: Geometry and Physics XII, Hainan (China), 10-14/03/2014

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:


. . . → 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