This only is denied to God: the power to undo the past.
by Agathon (448 BC - 400 BC)

Littlewood’s conjecture (1930)

Just learned about the following conjecture of Littlewood (1930), which looks very simple and which is apparently still open:

Let be two arbitrary real numbers. Denote by . Then

This conjecture is related to ergodic theory of It is not difficult to show that the set of pairs of number which donot satisfy . . . → Read More: Littlewood’s conjecture (1930)

Inverse of a unbounded operator ?

I’m stumbling over the problem of inversion of a unbounded operator on a Hilbert space. My operator is a perturbation of a Casimir operator acting by double differentiation of a tensor space. It ooks like an elliptic operator, so the inverse is expected to be (at least) bounded, with a “very reasonabe” bound. But . . . → Read More: Inverse of a unbounded operator ?

Rn-actions on n-dimensional manifolds (article in preparation)

Last updated 10/March/2012: This article is now finished. See

http://zung.zetamu.net/2012/03/geometry-of-rn-actions-on-n-manifolds-2012/

19/02/2012: We’re behind schedule now. Need to speed up !!!

This is the sketch of a research article in progress with a student of mine. I’m making the sketch of what we want to put in the article, and he takes care of writing . . . → Read More: Rn-actions on n-dimensional manifolds (article in preparation)

Rn-actions on n-dimensional manifolds

last updated: 18/jan/2012

This is a particular case of integrable non-Hamiltonian systems that my student Minh is working on with me for this thesis. We want to study such systems topologically. A real integrable system of type is nothing but a -action (generated by a family of commuting vector fields) on a -dimensional manifold.

. . . → Read More: Rn-actions on n-dimensional manifolds

A* journals in mathematics, according to Aussies

(updated 21/06/2013)

While looking for journals to submit my new preprints (I prefer to submit to places where I have not published before, in order to “collect” the journals :-)), I came across the following list of A* journals in mathematics, according to the Australian Mathematical Society. I don’t know which are their selection . . . → Read More: A* journals in mathematics, according to Aussies

Second-order models for asset prices

I’m doing this project about 2nd-order pricing models with a PhD student of mine. The project is quite ambitious. It aims to to be better than known models (Black-Scholes, stochastic volatility, 1st-order jump models, equilibrium, etc.) and be able to explain things that can’t be explained by previous models. Potential applications include investing & . . . → Read More: Second-order models for asset prices

Topology of integrable non-Hamiltonian systems

This is the research project of a PhD student of mine. Actually the general project is sufficiently large for a number of PhD theses. There are lots of open questions in the non-Hamiltonian case.

What my student is doing is to study simplest (mostly low-dimensional) cases, and with only nondegenerate singularities:

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

Geometric proof of Whitehead’s lemma ?

Whitehead’s lemma says that and where is a simple Lie algebra and is a linear representation of it. I know an algebraic proof which gives an explicit formula for the homotopy operator. I’m looking for a more geometric proof (using, for example, an averaging formula). Why ? Because the explicit homotopy operator in the . . . → Read More: Geometric proof of Whitehead’s lemma ?

Geometric approach to Levi decomposition

(A work in progress of mine)

Recently there have been very interesting papers by Crainic, Fernandes and company about geometric approaches to normal forms problems in Poisson geometry. Using groupoid techniques (in particular integration & the path method & their integrability iteria) they recover Conn’s linearization theorem (for Poisson structures), and also my linearization . . . → Read More: Geometric approach to Levi decomposition

1:1:2 resonance revisited

The aim of this quick note is to review Duistermaat’s formal nonintegrability result for 1:1:2 resonance.

We will use complex coordinates and write the quadratic part of the Hamiltonian function as

The monomes of degree 3 which commute with are:

Thus, a generic in Birkhoff normal forms will have 6 terms (is a linear . . . → Read More: 1:1:2 resonance revisited

1:2:2 resonance revisited

1:2:2 resonance was studied by Van der Aa and Verhulst who showed its asymptotic integrability, i.e. if H = H_2 + H_3 + …  is in Birkhoff normal form such that H_2 is in 1:2:2 resonance, then H_2 + H_3 is integrable. They also generalized the result to the n degrees of freedom 1:2:2:…:2 . . . → Read More: 1:2:2 resonance revisited

Formal non-integrability of resonant systems

I’m coming back here to an old but still open (as far as I know) problem:

Show that a generic Hamiltonian system with a fixed point and whose resonance degree at that point is at least 2 is not formally integrable.

Recall that, if the system is non-resonant then it’s formally integrable due to . . . → Read More: Formal non-integrability of resonant systems

Taylor-Green vortex

Taylor-Green vortex (1937) is a simple perioid initial value problem for the 3D Euler equation:

with

, ,

which leads to a complicated turbulent flow, and has become a standard example in fluid dynamics.

Taylor and Green conjectured that their vortex would develop a finite-time blowup. However, numerical simulations didn’t show a . . . → Read More: Taylor-Green vortex