by Leonid Polterovich and Felix Schlenk
In this essay we describe some of Dusa’s contributions to symplectic embedding problems.
Let us first say what these problems are about:
A symplectic manifold
Every symplectic manifold locally looks like
Denote by
The starting point of our discussion is the following problem, which demonstrates both rigid and flexible behaviour of symplectic mappings:
fill as much as possible of
Let
The inequalities
In the summer of 1991, both Dusa and
Leonid visited
Helmut Hofer in Bochum. Helmut suggested Leonid to work
on the concept of super-recurrence in Hamiltonian dynamics: indeed, symplectic maps tend to have
more periodic orbits than general volume preserving maps.
Leonid’s take on the problem was as follows: assume that for some
On a more personal note, I (Leonid) competed for Dusa’s attention with her son Thomas, at that time a young child who accompanied Dusa on her trip to Bochum. Dusa skillfully handled the case. Thus I got two private lessons for the price of one: not only in symplectic topology, but also in wise and very patient parenting. I use this occasion to express my gratitude and admiration to Dusa for both of them.
The above packing obstructions for
For symplectic packings of a ball by equal balls in higher dimensions
The papers
[1],
[2]
also contained the first step to the following algebraic reformulation of the general ball packing problem
(Volume constraint)
;(Constraint from exceptional spheres)
for every vector of nonnegative integers ; , that solves the Diophantine system and can be reduced to ; , by repeated Cremona moves.
Here, a Cremona move takes a vector
After a break of ten years, Dusa returned to symplectic embeddings in [6],
in which she studied the problem
This latter problem is encoded in the function
Define the Fibonacci stairs as the graph on
Theorem 2 (Fibonacci stairs, [8]):
On the interval
the function is given by the Fibonacci stairs.On the interval
we have except on nine disjoint intervals where is a step made from two segments. for all .
Thus
I (Felix) met Dusa for the first time twenty years ago at ETH Zürich, at the very beginning of my PhD Thesis. An year later I still did not know what to work on, but I had read a few papers, among them [3], where symplectic folding was invented. In Remark 2.4 therein the authors explained how this embedding method yields interesting lower bounds for
, and that the details will be published elsewhere. When Dusa visited ETH again, she said that I may work on this if I like. She gave me this problem just like a pebble, but for me it was a gem. My results looked safe, since it looked dangerous to use -curves in the orbifold obtained by removing the image of an ellipsoid in and collapsing the boundary. But in 2007, shortly after I had got a permanent position at ULB in Brussels, Dusa sent me a preprint of [6]. My thesis had gone up in smoke for a good part! I knew from scaling properties of symplectic capacities that her results imply the graph of for . Dusa immediately replied “Let’s work out together!”. The next weeks she patiently spent with explaining to me the methods. It then became clear that we need many computations, to see which solutions in are relevant. I was very happy about this, since computations I thought I could do. But Dusa computed about three times faster than me for several days. So I decided to use a computer code, something she may not be able to write. But at Brussels there was no Mathematica available (the only language I was familiar with), so after some days I took a train to Zurich, reactivated my old account, and then a day later sent Dusa a few pages of examples. I think it was the first and last time she was impressed by me, until she understood I had a code. We then soon guessed the right answer. Next Dusa generated a firework of ideas and methods; some failed, some worked but did not help, but some were just perfect, and so progress was steady and thriving. Showing that on the interval was the last hurdle, on which we got stuck for several months. One day Dusa sent me an intricate and overwhelming chain of estimates essentially settling the problem. When I asked her how she came up with this, she just wrote: “in sheer desperation!”
In conjunction with
Michael Hutchings’s capacities derived from his embedded contact homology,
associating with every open subset
More surprisingly, these methods also led to packing stability in higher dimensions:
Let
The ellipsoid embedding problem in dimensions
There are several reasons why we know much less about symplectic embeddings in dimension
positivity of intersections of
-holomorphic curves is an important tool in dimension four; -holomorphic curves in dimension four are much better understood, since often their existence comes from Taubes–Seiberg–Witten theory;symplectic inflation, that often can be used to show that the obstructions are sharp, can be generalized from four to higher dimensions, but to use this for constructing symplectic packings one now needs to show the existence of certain symplectic hypersurfaces, which for the time being seems out of reach.
Since ECH-capacities lead to much progress in dimension four, one can hope that other versions of
Floer
homology will provide interesting embedding obstructions also in higher dimensions
(see
([e8], Section 1.8.1)
for a discussion).
While ECH is isomorphic to a Seiberg–Witten Floer homology and thus intrinsically 4-dimensional,
mimicking the construction of ECH-capacities in terms of
Finally, let us mention that, in contrast to complex projective spaces, for certain target manifolds packings by
(not necessarily equal!) balls are completely flexible: a packing
exists provided it is not prohibited by the volume constraint. This
was established by
Janko Latschev,
Dusa McDuff and Felix Schlenk [9] for all
4-dimensional linear tori other than