Metaphysics is a dark ocean without shores or lighthouse, strewn with many a philosophic wreck.
by Immanuel Kant (1724-1804)

Tại sao vấn đề P vs NP khó vậy (2)

[Cưỡi ngựa xem hoa] [Bổ sung lần cuối: 10/Sep/2012]

Trong bài viết trước (Tại sao vấn đề P vs NP khó vậy) chưa hề nói đến các vũ khí đã dùng “để tấn công thành trì P|NP” . Bài này sẽ thử liệt kê dần các “vũ khí thông dụng” đã dung trong các . . . → Read More: Tại sao vấn đề P vs NP khó vậy (2)

Tại sao vấn đề P vs NP khó vậy ?

Đây chỉ là một bài cưỡi ngựa xem hoa thôi, vì sự hiểu biết của tôi về tin học chỉ ở mức “lớp 1″. Tuy nhiên, nghe người ta nói “vấn đề P vs NP” là vấn đề lý thuyết “quan trọng nhất của thời đại” nên cũng phải tìm hiểu nó xem sao, . . . → Read More: Tại sao vấn đề P vs NP khó vậy ?

Intrinsic convexity of almost-toric integrable Hamiltonian systems

This is a work in progress with Christophe Wacheux, a student of San.

Christophe went to see me in Toulouse to discuss about his thesis. After several discussions, I gave him 2 problems to work on. The first  is about semi-local classification of integrable Hamiltonian systems up to exact isomorphisms (i.e. diffeomorphisms which preserve . . . → Read More: Intrinsic convexity of almost-toric integrable Hamiltonian systems

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)

Decomposition of Dynamical Systems

arXiv:0903.4617

I’m interested in all kinds of decomposition of all kinds of systems.

I’m planning to write a paper on the decomposition of dynamical systems into fundamental states. This project is a bit vague. I want to make some ideas/results more precise and easier to apply before writing things up.

By a “dynamical system” . . . → Read More: Decomposition of Dynamical Systems

Geometry of integrable vector fields on surfaces

Updated: 04/April/2012

This paper is on schedule. It is almost finished now, and has about 30 pages. To be put on arxiv tomorrow.

Updated: 19/03/2012

Section: Generic nilpotent singularities

(X,F) where F regular, X nilpotent:

F =F(y),

X = y d/dx + …

Since X(F) = 0 –> X = (y + …) d/dx

. . . → Read More: Geometry of integrable vector fields on surfaces

Nondegeneracy of simple Lie algebras of real rank 1

This is work in progress with Ph. Monnier.

The nondegeneracy here is in the sense of Weinstein: a real Lie algebra g is called smoothly nondegenerate if any smooth Poisson structure whose linear part is g* is locally smoothly linearizable (i.e. isomorphic to g*)

The theorem that we want to prove gives a complete . . . → Read More: Nondegeneracy of simple Lie algebras of real rank 1

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

Smooth linearization of sl(2,R) and SL(2,R) actions ?

This is a particular case of a larger problem of linearization of Lie group and Lie algebra actions. The case of sl(2,R) and SL(2,R) is already non-trivial.

Guillemin and Sternberg gave an example of a smooth sl(2,R) action on R3 which is not linearizable.

Cairns and Ghys gave an example of a smooth SL(2,R) . . . → Read More: Smooth linearization of sl(2,R) and SL(2,R) actions ?

Applications of Nash-Moser normal form theorem ?

Project with Eva Miranda and Philippe Monnier

We have invented a “hammer” called “abstract Nash-Moser normal form theorem” and now are looking for “nails”

We used our hammer for the problem of rigidity of Hamiltonian actions on Poisson manifolds (see paper in Advances Math 2012). We don’t know yet to what problems (beyond . . . → Read More: Applications of Nash-Moser normal form theorem ?

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 ?