Category Archives: Open questions

Mean action and the Calabi invariant

I recently posted a new paper, “Mean action and the Calabi invariant“. There is a bit of a story about where this paper comes from; I didn’t try to explain this in the paper, in order to keep things focused, … Continue reading

Posted in ECH, Open questions | 1 Comment

Is cylindrical contact homology defined with integer coefficients?

A confusing issue about cylindrical contact homology is whether it is defined with integer or rational coefficients. I would now like to try to clear this up once and for all. If I am not mistaken, the conclusions are the … Continue reading

Posted in Contact homology, Open questions | 7 Comments

A guest post by Dans C-G and P

[The following is a guest post by Dan Cristofaro-Gardiner and Dan Pomerleano. If anyone else is interested in contributing a guest post, please feel free to contact me. A blog is a good outlet for short or informal mathematical thoughts … Continue reading

Posted in ECH, Open questions | 1 Comment

An obstruction to cylindrical contact homology?

Last week I was at a very nice conference at IMPA, and one of the topics of informal discussion was to what extent cylindrical contact homology can be defined, in the absence of contractible Reeb orbits, but without using polyfolds. … Continue reading

Posted in Contact homology, Open questions | 7 Comments

Update on the short Reeb orbit conjecture

In an earlier post, I stated a conjecture that every contact three-manifold has a Reeb orbit with an upper bound on its symplectic action in terms of the volume of the contact manifold. There hasn’t been much progress on this … Continue reading

Posted in Open questions | 4 Comments

Do we really need Seiberg-Witten theory to understand symplectic embeddings of 4d ellipsoids?

Dusa McDuff recently proved that the interior of the 4-dimensional ellipsoid symplectically embeds into if and only if .  Here, if and are positive real numbers, then denotes the sequence obtained by taking all nonnegative linear combinations of and and … Continue reading

Posted in ECH, Open questions | 2 Comments

Fun with ECH capacities 2

This is a continuation of the previous post about computing some examples of ECH capacities.  My goals in this post are (1) to recap some of the previous discussion using better notation, (2) to explain an effective way to compute … Continue reading