2,022 reputation
818
bio website homepages.ulb.ac.be/~samulisi
location Nantes, France
age
visits member for 3 years, 9 months
seen Dec 1 at 5:00

Postdoc working in symplectic/contact topology. I'm currently at the Université de Nantes.


May
7
answered An alternative description of the first Stiefel-Whitney class
May
4
comment When does a vector field admit orthogonal fields?
@Yvoz How does this last observation of yours not answer your question? It seems like a characterization of when such a vector field exists. What are you looking for if that isn't the answer?
May
3
comment When does a vector field admit orthogonal fields?
Observe that you don't need orthogonality for the second vector field -- you just need pointwise linearly independent and nowhere vanishing. Then, since $\mathbb{R}^3$ is oriented, you get a third vector field free. You are therefore trying to find a trivialization of the tangent bundle of the quotient of $U$ by the flow of $X$. In general, this quotient will be a horrible space and won't have a tangent bundle, but maybe something can be said anyway. (Note that in your two examples, the quotients are $\mathbb{R}^2$ are $S^2$ respectively, and in particular very nice.)
May
3
comment An alternative description of the first Stiefel-Whitney class
I was not thinking about Steenrod squares etc. I really was thinking about their chapter on obstruction theory, Chapter 12. I guess M&S refer back to Steenrod's book, which I haven't studied as carefully as I should have. On page 143, they explain how you can reconstruct the obstruction classes from the $w_i$, at least for small $i$.
May
3
comment An alternative description of the first Stiefel-Whitney class
@Mehdi : yes, that's what I was thinking about (to your first comment). I'll try to write something soon, but the short version of what I am thinking about is that the $\mathbb{Z}/2$ that shows up in the sequence $1 \to SO(n) \to O(n) \to \mathbb{Z}/2 \to 1$.
May
3
comment An alternative description of the first Stiefel-Whitney class
What definition(s) of $w_1$ do you know? If you have been studying Milnor-Stasheff, take a look at the obstruction theory chapter.
May
1
answered Question about symplectic tranformations
May
1
comment An other question about Theorem 3.1 from Morse theory by Milnor
let us continue this discussion in chat
May
1
comment An other question about Theorem 3.1 from Morse theory by Milnor
Do you understand what $\phi_t$ is?
Apr
30
comment An other question about Theorem 3.1 from Morse theory by Milnor
The tricky part is that $\phi_{b-a}$ maps $M^a$ to $M^b$. To check this, calculate what $f( \phi_{b-a}(x))$ is, if $f(x) \le a$. This is where you use the fact that your new vector field flows downwards so that the time $t$ flow changes the value of $f$ by $t$.
Apr
30
comment An other question about Theorem 3.1 from Morse theory by Milnor
The fact that it is a diffeomorphism onto its image follows just from the existence/uniqueness of ODE, the long-time existence (essentially by construction) and smoothness with respect to initial condition. I.e. this is standard ODE theory.
Apr
30
comment An other question about Theorem 3.1 from Morse theory by Milnor
The purpose of (1) is to make (2) obvious and to make it easy to write down a formula for $r_t$. Can you construct a simple example (e.g. for domains in $\mathbb{R}^2$) and see what he is doing? Can you say more about where you are stuck?
Apr
30
answered Implicit Function Theorem and Rank Theorem Misunderstandings.
Apr
30
comment Dynamical Systems. Bendixson's and Dulac-Bendixson's theorems.
Great timing, I was just typing essentially the same thing. While this is clear from what you wrote, I want to emphasise that $U$ has positive area because $C$ is embedded. ($C$ is embedded by uniqueness of solutions to ODE.)
Apr
27
answered Non-degenerate solutions to constant Hamiltonian flow
Apr
27
answered Lagrangian subspaces
Apr
16
comment Area of flux homomorphism in symplectic topology
I don't understand what you are trying to do, specifically. Are you trying to compute this explicitly? for that to make sense, it seems to me that you need some more explicit information. Or are you trying to do something else? this isn't clear to me.
Apr
13
comment Structure on manifolds
I should add the question: have you studied surfaces in $\mathbb{R}^3$? This is the easiest scenario with a shape operator, and it might be worthwhile to understand this first. If you already know about this use of the shape operator, I can't tell you anything further.
Apr
13
comment Structure on manifolds
@Merri: I don't know the book you mention, and I know very little about uses of shape operators. However, I would guess that the key feature of a "shape operator" is that it is the differential of a "Gauss map". There are many situations in which you have something that looks like a Gauss map (but isn't the classical one), so maybe there are many non-standard shape operators. I don't really know.
Apr
11
comment Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
let us continue this discussion in chat