My religion is very simple. My religion is kindness.
by Dalai Lama

Entropy of geometric structures

Updated 06/Dec/2011: This paper is now published online in Bulletin Brazilian Math Soc, Volume 42, Number 4, 853-867:

http://www.springerlink.com/content/t623n03m400p6870/

Finally I had the courage to revise my paper on the entropy of geometric structures. The revised version is here (pdf file).

Arxiv: http://arxiv.org/abs/1109.5249

This paper is relatively simple, but I still like it very . . . → Read More: Entropy of geometric structures

Rigidity of Hamiltonian actions (revised version)

Updated 21/Oct/2011: The editors just informed us that this paper has been officially accepted for publication in Advances Math.

We have revised our paper (with Eva Miranda and Philippe Monnier) on the rigidity of Hamiltonian actions of compact semisimple Lie groups on Poisson manifolds.

The revised version is available here.

This paper was submitted . . . → Read More: Rigidity of Hamiltonian actions (revised version)

Eight Vietnamese win Hellman/Hammett Human Rights Award

Source: Human Rights Watch

(Bangkok) – Eight Vietnamese writers are among a diverse group of 48 writers from 24 countries who have received the prestigious Hellman/Hammett award recognizing writers who demonstrate courage and conviction in the face of political persecution, Human Rights Watch said today [14/Sep/2011]

“Vietnamese writers are frequently threatened, assaulted, or even . . . → Read More: Eight Vietnamese win Hellman/Hammett Human Rights Award

List of books to be bought for the school on financial mathematics

We want to buy some books on financial mathematics and bring them to Vietnam for the thematic school in Doson (24/Oct-01/Nov/2011).After the school, these books will be donated to a library in Hanoi.

I need  a list of  best books to buy. If you have any suggestion, please let me know by writing a . . . → Read More: List of books to be bought for the school on financial mathematics

Lectures on Poisson Geometry (Geometry and Topology Monographs, Vol. 17)

Finally, “Lectures on Poisson Geometry”, which is a collection of lectures on various aspects of Poisson geometry, edited by T. Ratiu, A. Weinstein, and myself, has appeared as Volume 17 of the series Geometry and Topology Monographs:

http://pjm.math.berkeley.edu/gtm/2011/17/

Poisson geometry is a rapidly growing subject, with many interactions and applications in areas of mathematics . . . → Read More: Lectures on Poisson Geometry (Geometry and Topology Monographs, Vol. 17)

Nondegenerate singularities of integrable non-Hamiltonian systems

Last updated: 07/Apr/2011

The purpose of this note is to study nondegenerate singularities of integrable non-Hamiltonian systems. In particular we want to extend the Vey-Eliasson theorem about the local linearization of nondegenerate singularities of integrable Hamiltonian systems to the non-Hamiltonian case, and show that, in the non-Hamiltonian case, nondegenerate singularities are also rigid and . . . → Read More: Nondegenerate singularities of integrable non-Hamiltonian systems

Topology of integrable non-Hamiltonian systems

Last updated: 01/Apr/2011

This is a research project on which a PhD student of mine is working with me. Please don’t steal the ideas and results that I discuss here.

The problem is to study the topology and geometry of proper non-Hamiltonian integrable dynamical systems on manifold. A non-Hamiltonian integrable system consists of:

* . . . → Read More: Topology of integrable non-Hamiltonian systems

Hidden symmetries of mathematical objects

A general philosophy is that, mathematical objects have symmetry groups, and can be classified by these groups. The Galois theory is an example. Transformation groups or groupoids, linear representation theory, classification of metrics by holonomy groups, etc.,  are also instances of this philosophy.

There are objects, which a-priori have no symmetries, but still have . . . → Read More: Hidden symmetries of mathematical objects

Solar Power Industry Analysis

(This is a report that I made for a seminar on financial mathematics and investing, and a small investment fund created by the participants of this seminar. This report is not a recommendation to buy or sell anything to anyone else).

Key findings:

* Solar energy is the best source of energy: abundant, free, . . . → Read More: Solar Power Industry Analysis

Rigidity of Hamiltonian actions

Finally (after several years of dragging our feet), my colleagues Eva Miranda and Philippe Monnier and I have just finished our paper on the rigidity of Hamiltonian actions of compact semisimple Lie groups on Poisson manifolds.

The rigidity phenomenon here is quite natural. It has been  known for a long time that compact group . . . → Read More: Rigidity of Hamiltonian actions

The paper on entropy of geometric structures is finally done!

I’m so glad to have finished my little paper on entropy of geometric structures today, just before the deadline (which is the last day of 2010).

The paper can be seen here in PDF format: Entropy of Geometric Structures.

Abstract: We give a notion of entropy for general gemetric structures, which generalizes well-known notions . . . → Read More: The paper on entropy of geometric structures is finally done!

Entropy of geometric structures

Last updated: 22/Oct/2010

(This is the first draft of a note to be submitted to a special issue of the Bulletin of the Brazilian Mathematical Society)

Geometric structures

In this note, by a geometric structure, we mean a normed vector bundle $A \to M$ over a vector bundle (i.e. on each fiber there is . . . → Read More: Entropy of geometric structures

Notes on INS (14): CKN theory

Last updated: 27/Oct/2010

CKN stands for Caffarelli-Kohn-Nirenberg. The theory is about partial regularity of solutions of INS. One should probably add the name of Scheffer, who introduced the concepts that CKN improved/generalized.

The main result is that the (parabolic) 1-dimensional Hausdorff measure of the (hypothetical) singular set in space-time is zero (which means that, . . . → Read More: Notes on INS (14): CKN theory