Bây giờ trung thực phải đặt lên trước vì là cái thiếu nhất!
by Cố Thủ tướng Phạm Văn Đồng

Zucon flows

I’m constructing here an example of what I want to call “a zucon flow” of incompressible fluid. It is not a solution of anything (or more precisely,  you’ll need an appropriate external force to get such a flow). But nevertheless it’s interesting to imagine such flows.

Zucon means “petit voyou”, and it has an . . . → Read More: Zucon flows

Non-uniqueness of weak solutions to Euler equations (Notes on INS)

There are examples, due to Shnirelman and Scheffer,  of non-zero weak solutions to the Euler equations (without external forces), which are zero when and . That is, a movement which appears from no-where, and then disappears after some finite time. Sounds a bit crazy, doesn’t it ?

The method used by Shnirelman is to . . . → Read More: Non-uniqueness of weak solutions to Euler equations (Notes on INS)

Gibbs measures

These are the notes that I’m taking for myself in order to learn some statistical mechanics.

The main reference is: Anton Bovier, Lecture notes on Gibbs measures and phase transitions (Bonn University)

In thermodynamics one has:

where are respectively the mechanical, chemical and thermal components of the energy.

is also denoted by ,

. . . → Read More: Gibbs measures

Understanding the formula dQ = T dS

I’m still trying to understand the formula dQ= T dS, which goes back to Clausius (mid 19th century). It can be also written as dQ/dS = T, or dS = dQ/T. Here T is the absolute temperature, S is the entropy, and Q is the heat (thermal energy). This formula is often taken “for . . . → Read More: Understanding the formula dQ = T dS

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

Some rigidity stuff

While looking for results on rigidity, I stumbled upon the following paper by Fisher:

David Fisher, First cohomology and local rigidity of group actions, Ann. Math. (to appear ? a preprint is available since 2009)

The main result of this paper is:

Thm: Let \Gamma be a finitedly presented group, (M,g) a compact Riemannian . . . → Read More: Some rigidity stuff

Second order models

Here’s a real research problem that I’m trying to think about:

How to introduce and study second order models (say for stochastic processes and financial time series) ?

By second order, I mean stochastic difference or diffenrential equations of second order (as opposed to 1st order used in existing literature).

Why 2nd order ?

. . . → Read More: Second order models

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

Geometry Conference, Hanoi 18-22/April/2011

http://www-ma1.upc.es/~miranda/hanoi/conference/

Geometrical methods in Dynamics and Topology

Hanoi National University of Education, Hanoi, 18-22/April/2011

This is an international mathematical conference on the occasion of the 65th anniversary of  Hanoi National University of Education.

Main speakers include:

Marc Chaperon (Paris VII)

Alain Chenciner (Observatoire de Paris-IMCCE)

Jesus Gonzalo (Madrid)

Basak Gurel (Vanderbilt)

Mark Hamilton* ( . . . → Read More: Geometry Conference, Hanoi 18-22/April/2011

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

Local normal forms, differential Galois, resonances, and formal non-integrability

I’m not through with “fundamental maths” yet, so I’ll have to cook up a new paper in 2011 :-)

Same old things: normal forms, Galois theory, resonances, and non-integrability. Some new twists: the Galois group here will be a local one (defined in the neighborhood of a fixed point, and not a non-stationary solution). . . . → Read More: Local normal forms, differential Galois, resonances, and formal non-integrability

External movement commutes with internal movement

There is a very interesting general principle of mechanics (which is not exactly new, but which I have not seen formally written anywhere this way):

External movement commutes with internal movement.

For example, consider a person in a plane which flies. When the person moves in the plane, it’s an internal movement with respect . . . → Read More: External movement commutes with internal movement