# Sam Lisi

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website        homepages.ulb.ac.be/~samulisi
location       Nantes, France
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member for     2 years, 2 months
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Postdoc working in symplectic/contact topology. I'm currently at the Université de Nantes.

|            |                  | bio            | visits                        |            |                   |
|------------|------------------|----------------|-------------------------------|------------|-------------------|
|            | 1,617 reputation | website        | homepages.ulb.ac.be/~samulisi | member for | 2 years, 2 months |
| 515 badges | location         | Nantes, France | seen                          | 1 hour ago |                   |

# 219 Actions

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| May1  | comment  | An other question about Theorem 3.1 from Morse theory by Milnor                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| May1  | comment  | An other question about Theorem 3.1 from Morse theory by MilnorDo you understand what $\phi_t$ is?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
| Apr30 | comment  | An other question about Theorem 3.1 from Morse theory by MilnorThe 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 $t$.                                                                                                                                                                                                                                                                                                                |
| Apr30 | comment  | An other question about Theorem 3.1 from Morse theory by MilnorThe 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.                                                                                                                                                                                                                                                                                                                                     |
| Apr30 | comment  | An other question about Theorem 3.1 from Morse theory by MilnorThe 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?                                                                                                                                                                                                                                                                                                                                       |
| Apr30 | answered | Implicit Function Theorem and Rank Theorem Misunderstandings.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| Apr30 | 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.)                                                                                                                                                                                                                                                                                                                                                     |
| Apr27 | answered | Non-degenerate solutions to constant Hamiltonian flow                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| Apr27 | comment  | Hamiltonian Isotopy in Symplectic geometryCan you give a bit more detail about what you are looking for? What do you mean by "visualize"? what makes a context "good"? Have you studied any classical physics where the Hamiltonian is giving the total energy of a conservative system? if so, this is the right visualization (at least on cotangent bundles).                                                                                                                                                                                                                                                                                    |
| Apr27 | answered | Lagrangian subspaces                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| Apr16 | comment  | Area of flux homomorphism in symplectic topologyI 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.                                                                                                                                                                                                                                                                                                                                         |
| Apr13 | comment  | global vector fields in local coordinatesIn general you can't, for the reason you explained. What are you trying to do?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| Apr13 | comment  | Structure on manifoldsI 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.                                                                                                                                                                                                                                                                                                                                                 |
| Apr13 | 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.                                                                                                                                                                                                               |
| Apr11 | comment  | Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| Apr11 | comment  | Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?I think you have some typos in your proposed modification of Theorem 3, or else I have no idea what you are asking for. In particular, I think you mean $f(E)$ instead of $F(E)$, but I still don't understand what exactly you are trying to accomplish.                                                                                                                                                                                                                                                                                              |
| Apr11 | comment  | Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?Observe that if the initial data is contained in the interior of the orthant ($\Omega$ above), then, for small time, the solution curve stays in $\Omega$, and thus you can use the standard chain rule. What's not at all clear is what behaviour you want to see at the boundary.                                                                                                                                                                                                                                                                    |
| Apr11 | comment  | Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?I don't understand your motivation. Call $\Omega$ the (strictly) positive orthant. Then the non-negative orthant is $\overline \Omega$. If your initial data $x_0 \in \Omega$, then for small time, your solution curve $x(t)$ will live in $\Omega$. However, if you have initial data at the boundary $x_0 \in \partial \overline{\Omega}$, there is no reason for your integral curve to stay in $\overline \Omega$ even infinitesimally. In the former case, one expects $V(x(t))$ to make sense for short time, whereas in the latter, you don't. |
| Apr11 | comment  | Symplectic geometry as a prequisite for Heegaard Floer homologyDo you have someone in your department who can give you some guidance? there is a lot of symplectic geometry in both of these books that is not relevant to you and there are a number of technical points about holomorphic curves that are not explained in these books. A correct answer to your question, imo, requires more information about what you know and about what you are interested in doing. Unfortunately, even with this information, I wouldn't be qualified to give you a good answer, since I don't work in Heegaard Floer homology.                            |
| Apr8  | comment  | Why is Cartan Formula just an avatar of Leibniz rule?Thank you for the bounty. I am grateful to you for the video.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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