3. Postdoctoral Program - MSRI

Ozsváth–Szabó Heegaard homology theory and Hutchings–Taubes Embedded Contact homology
(ECH) theory. Two groups of researchers successfully explored two different approaches
to this problem, and this led to further developments within the program.
Taubes and Hutchings completed their work relating Seiberg–Witten–Floer theory to ECH.
This has very important consequences, including a proof of the Arnold chord conjecture in 3
dimensions. In another fundamental step, Taubes found a way to establish the long sought
equivalence between Seiberg–Witten–Floer theory and Ozsváth–Szabó theory, by building a
direct geometric link between the ECH complex and the Ozsváth–Szabó complex. He is working
out the details of this beautiful idea jointly with Yi-Jen Lee and postdoc Cagatay Kutluhan.
A survey paper, and the first long installment of the proof, are now on the web.
At the same time, Colin, Ghiggini and Honda found a direct proof of the equivalence between
Heegaard Floer homology and ECH. They use an entirely different approach, involving Giroux’
open book decompositions contact 3-manifolods. This is still work in progress, but the first
paper in a projected series has again been posted on the web.
Applications to embedding problems. The work of Taubes and Hutchings interacted productively
with other developments in the program. Throughout the Spring semester, there was
much discussion of the state of symplectic embedding problems in dimension 4. Several breakthroughs
had been made just before, by Hind–Kerman and by McDuff–Schlenk. This prompted
Hutchings to develop his ideas on ECH capacities sufficiently far for McDuff to prove the Hofer
conjecture on embeddings of ellipses. Hind also introduced new ideas about embeddings of
polydiscs, which turned out to be related to work of Fukaya–Oh–Ohta–Ono on displacing tori.
Symplectic Field Theory, Contact Homology and Applications. Bourgeois, Ekholm and Eliashberg
developed a Legendrian surgery exact triangle for computing contact and symplectic
homology, as well as some other symplectic invariants. In particular, this gave an explicit
formula for the symplectic homology complex in terms of the Legendrian homology algebra
of the attaching spheres. As an application, Postdoc Maydanskiy and UC Berkeley graduate
student Ganatra showed that this implies the Seidel conjecture for symplectic homology of a
manifold described as a Lefschetz fibration. As a consequence, concrete progress was made
in constructing new exotic symplectic structures: for instance, Abouzaid and Seidel showed
that any complex affine variety of sufficiently high dimension admits infinitely many convex at
infinity distinct symplectic structures, not distinguished by classical homotopy theory. They,
and independently postdoc McLean, also showed that R 2n , n ≥ 6, admits uncountably many
distinct convex at inifinity symplectic structures (an analogue of the celebrated theorem of
Gompf about differentiable structures on R 4 ). Moreover, McLean showed that the problem of
classifying Weinstein-type symplectic structures on T ∗ S n , for n ≥ 7, is algorithmically unsolvable.
The corresponding question for R 2n is still open, but now looks ripe to be attacked. All
this is fundamentally driven by advances in computing SFT invariants. Initiating the next step
in these developments, very recent work of Bourgeois, Ekholm and Eliashberg finds a formula
for the symplectic homology product, and some other invariants, in terms of the Legendrian
homology algebra.
Several years ago Ng constructed combinatorial invariants of knots in R 3 that have proven to be
very effective invariants and surprisingly related to many very different classical knot invariants.
Ekholm, Etnyre, Ng and Sullivan mostly completed work showing that Ng’s invariant is really
the contact homology of the conomral lift of a knot to the unit cotangent bundle of R 3 . More
strikingly they showed how to lift a contact structure to the unit conormal bundle so that
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