
By NTZung, on May 2nd, 2011%
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)
By NTZung, on April 11th, 2011%
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
By NTZung, on April 11th, 2011%
There are examples, due to Shnirelman and Scheffer, of nonzero weak solutions to the Euler equations (without external forces), which are zero when and . That is, a movement which appears from nowhere, and then disappears after some finite time. Sounds a bit crazy, doesn’t it ?
The method used by Shnirelman is to . . . → Read More: Nonuniqueness of weak solutions to Euler equations (Notes on INS)
By NTZung, on April 9th, 2011%
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
By NTZung, on April 8th, 2011%
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
By NTZung, on April 4th, 2011%
Last updated: 07/Apr/2011
The purpose of this note is to study nondegenerate singularities of integrable nonHamiltonian systems. In particular we want to extend the VeyEliasson theorem about the local linearization of nondegenerate singularities of integrable Hamiltonian systems to the nonHamiltonian case, and show that, in the nonHamiltonian case, nondegenerate singularities are also rigid and . . . → Read More: Nondegenerate singularities of integrable nonHamiltonian systems
By NTZung, on March 31st, 2011%
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 nonHamiltonian integrable dynamical systems on manifold. A nonHamiltonian integrable system consists of:
* . . . → Read More: Topology of integrable nonHamiltonian systems
By NTZung, on March 28th, 2011%
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
By NTZung, on March 28th, 2011%
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
By NTZung, on March 17th, 2011%
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 apriori have no symmetries, but still have . . . → Read More: Hidden symmetries of mathematical objects
By NTZung, on February 8th, 2011%
http://wwwma1.upc.es/~miranda/hanoi/conference/
Geometrical methods in Dynamics and Topology
Hanoi National University of Education, Hanoi, 1822/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 ParisIMCCE)
Jesus Gonzalo (Madrid)
Basak Gurel (Vanderbilt)
Mark Hamilton* ( . . . → Read More: Geometry Conference, Hanoi 1822/April/2011
By NTZung, on February 1st, 2011%
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
By NTZung, on January 1st, 2011%
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 nonintegrability. Some new twists: the Galois group here will be a local one (defined in the neighborhood of a fixed point, and not a nonstationary solution). . . . → Read More: Local normal forms, differential Galois, resonances, and formal nonintegrability


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