User chris woodward - MathOverflow

Answer by Chris Woodward for Intuitive explanation for the use of matrix factorizations in knot theory

2010-12-22T03:35:07Z

Many knot homologies are expected to have Floer-theoretic interpretations. However, in Floer theory often the chain "complex" $CF(L_0,L_1)$ is not a complex but rather an a-infty bimodule over a pair of curved a-infty algebras; matrix factorizations are more or less a special case of these, where the curvature of the ainfty algebras is a multiple of the identity. Under some nice (monotone or exact) assumptions $CF(L,L)$ has differential that squares to zero, since the curvatures of the a-infty algebras on both sides occur with opposite sign and so cancel if the Lagrangians are the same (or related by a symplectomorphism). But even in the monotone or exact situations $CF(L_0,L_1)$ can have a differential which gives a matrix factorization. Now Manolescu has proposed a symplectic interpretation of Khovanov-Rozansky for links, which looks like $CF(L,\beta(L))$ where \beta is a braid presentation of the link, but so far there doesn't seem to be a proposal for graphs. Probably for graphs the invariant would be the homotopy type of $CF(L_0,L_1)$ (or some more general quilted Floer chain group) which under suitable monotonicity assumptions will be a matrix factorization. Note that Kamnitzer has a proposal for a Floer-theoretic version for arbitrary G; it would be interesting to see if one could extend this proposal to the graph case, which one would probably need for an exact triangle.

Floer's space closed under products?

2010-08-20T14:34:48Z

Floer (in "The unregularized gradient flow of the symplectic action") defined a dense subspace of $C^\infty$ with the structure of a Banach space, with norm $\Vert f \Vert = \sum_{k \ge 0} \epsilon_k \Vert f \Vert_{C^k}$ for some constants $\epsilon_k$.

Question: How do the constants $\epsilon_k$ in Floer's norm behave as $k \to \infty$? He just shows that some constants exist for which the resulting space is dense in $C^\infty$. Do the constants go to infinity, to zero, or neither?

Subquestion: Is Floer's space closed under products of functions?

Answer by Chris Woodward for Refereeing a Paper

2010-08-24T23:29:24Z

I've been spending a fair amount of time editing a journal this year, and it's pretty amazing what different people think of as a "referee report". The first thing you should keep in mind is that the editors will be incredibly appreciative if you look at the paper in detail and send in comments in a timely manner, whatever the comments are. In my mind a good referee report begins with a very short (a few sentences) summary of the result and the argument. It includes an opinion on whether the result and proof are (i) correct (ii) readable (iii) interesting to lots or only a few people; also (iv) a recommendation on whether it is good enough to appear in the given journal, or alternatively/also comparisons to the quality of one or two other recent papers in the field, or just a statement that the referee isn't sure if it is good enough or not, and (v) a non-empty list of specific corrections/suggestions.
Re (3-4) it's perfectly fine to send in a one-line request that the paper be revised so that it is written in correct English. It's not your job to correct grammatical mistakes if there are more than a few. Re (5-6) if an argument is hard to follow, you can just ask for a revision in which the authors explain more. As an editor, it's quite easy (with computers and e-mail being what they are) to request a revision, even after the referee has only read part of the paper.

Short-time Existence/Uniqueness for Non-linear Schrodinger with Loss of Several Derivatives

2010-08-20T14:28:27Z

I have the following question about short-time existence and uniqueness results for non-linear schrodinger equations (NLSE) where the non-linearity involves a loss of derivatives (in my case, it is a non-local non-linearity involving a loss of two derivatives.) It seems that most current techniques allow some small number of derivatives in the non-linearity, except for a series of papers by Poppenberg, which use Nash-Moser. (Unfortunately he seems to have left math.)

Question: are there any short-time existence and uniqueness results for NLSE with loss of several derivatives for a space of initial conditions smaller than $C^\infty$? I have in mind for example a space introduced by Floer which is a dense subspace of $C^\infty$ that carries the structure of a Banach space whose norm is a combination of all C^k norms. One could imagine doing something similar by combining all Sobolev norms. I would be quite happy with a short-time existence and uniqueness result for initial conditions in some dense subspace of C^\infty.

p.s. I am posting in a second a related question about Floer's Banach space with a symplectic geometry tag.

Answer by Chris Woodward for What's the chain level Gromov-Witten theory

2010-07-31T01:44:15Z

A theory of the type you might be talking about is stated in Sullivan's "Theorem 4" in "String topology and sigma models", http://books.google.com/books?hl=en&lr=&id=yuGic0WClQ4C&oi=fnd&pg=PA1&dq=string+topology+and+sigma+models+sullivan&ots=2Oympt_Z4g&sig=W4GOnI6Cn-7IjsJEH7x9c8-qnnY#v=onepage&q=string%20topology%20and%20sigma%20models%20sullivan&f=false. Whether it is a Theorem is somewhat unclear to me.

Hamiltonian displaceability of tori in symplectic balls

2010-05-21T00:17:27Z

Here is my first try at a question, which is a really easy to state question about displaceability:

Let $D$ be the unit disk in the complex plane $D = { |z| \leq 1 }$ equipped with its standard symplectic form and for $r \in (0,1)$ let $S(r) \subset D$ be the Lagrangian circle of radius $r$ centered at $0$, enclosing area $\pi r^2$.

Question: for which $r_1,\ldots, r_n$ is the Lagrangian torus $S(r_1) \times \ldots \times S(r_n)$ Hamiltonian displaceable in $D^n$?

Conjecture: for a given $r_1,\ldots, r_n$, the Lagrangian is non-displaceable iff $r_j^2 \ge 1/2$ for all $j$.

Known: Using McDuff's probes, one can see that if some $r_j^2 < 1/2$, then the Lagrangian is displaceable. I think I can show that if each $r_j^2$ lies in the set ${ 1/2, 2/3,3/4, \ldots }$ then the Lagrangian is non-displaceable, by embedding D^n in a product of weighted projective lines and showing the Floer homology of the Lagrangian is non-vanishing. But there must be a way of doing better.

Sub-question: (which came out of a discussion with Abouzaid): is there a Floer-theoretic way of seeing the non-displaceability of $S(r)$ for $r^2 \ge 1/2$? This is easy to prove by elementary means.

Comment by Chris Woodward

2012-11-19T00:10:16Z

By the way, just to correct the earlier comment, there was a Floridia-Syracuse railway at the time Orsi was writing. You can see its path here: discusmedia.com/catalog.php?id=25008, and the list of stations at it.wikipedia.org/wiki/… More fun trivia!

Comment by Chris Woodward

2012-10-20T17:38:50Z

Agree, and thanks for the link. I wouldn't want to say that the shopping center is the probable location, but it seems to me based on what I know so far that the area around it has the maximum probability density.

Comment by Chris Woodward

2010-08-25T12:40:06Z

All comments are helpful, there is just a lot of variation. If some is senior, then they will have their own style of referee report and I wouldn't feel comfortable asking them to fill something like a form. Anyway most of the editorial work seems to be dealing with papers that people seem to think are good but no one actually wants to read, and so my attention to individualizing request letters often wavers. If the world was full of people like RBega .....

Comment by Chris Woodward

2010-08-20T21:48:58Z

(ctd) So it seems my problem is closest to (3) in Piero's answer, and in fact it seems what I need to do is try to find for which Gevrey class of initial conditions is good. Do you have a recommended reference involving Gevrey initial conditions with loss of derivatives in non-linear Schrodinger? I would be very happy to show uniqueness for $G^s$ initial condition for some $s > 1$.

Comment by Chris Woodward

2010-08-20T21:48:48Z

Thanks Piero and Deane! (1) In my particular equation the potential is obtained from an expression involving the fourth derivatives of the square of the wave-function, by solving a second-order Poisson equation. So if $\psi$ is in $H^s$, then $V(\psi)$ is in $H^{s-2}$ for s sufficiently large, So in this sense $k=2$. But $V(\psi)$ depends non-locally on $\psi$ in much the same sense as in the paper by Ginibre-Velo on non-local interaction in non-linear Schrodinger.