[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)
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 ?
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
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)
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
This paper is on schedule. It is almost finished now, and has about 30 pages. To be put on arxiv tomorrow.
Section: Generic nilpotent singularities
(X,F) where F regular, X nilpotent:
X = y d/dx + …
Since X(F) = 0 –> X = (y + …) d/dx
. . . → Read More: Geometry of integrable vector fields on surfaces
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
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
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 ?
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 ?
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
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
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 ?