3. Postdoctoral Program - MSRI

Mark spend his time at MSRI working on two papers on symplectic homology. The first paper
”A Spectral sequence for symplectic homology” constructs a spectral sequence converging to
symplectic homology of a Lefschetz fibration whose E 1 pages are Floer homology groups of the
monodromy symplectomorphism of this Lefschetz fibration; this is then used to prove a theorem
about fixed points of certain symplectomorphisms. The second paper, ” The growth rate of
symplectic homology and applications” prove several properties of an invariant of Liouville d
omains called the growth rate of symplectic homology. Mark uses growth rates to show that
the unit cotangent bundle of a rationally hyperbolic manifold is not Stein fillable by a smooth
affine variety. Mark also has a growth rate criterion for infinitely many Reeb orbits, and a
sketch of a computability result which will be written up in a third paper.
Sikimeti Ma’u
PhD: Rutgers University, 2008
Position prior to MSRI membership: postdoc at MIT
Position after MSRI membership: NSF postdoc at Barnard College
Mentor: Denis Auroux, Eleny Ionel, Dusa McDuff
While at MSRI Sikimeti worked on analytical and algebraic aspects of Quilted Floer theory.
During the Fall 2009 she completed ”Gluing Pseudoholomorphic Quilted Disks”, and in the
spring 2010 she started ”Quilted strips, graph associahedra, and A-infinity n-modules” (completed)
and ”A-infinity bimodules for Lagrangian correspondences” (near completion).
Sikimeti mentioned that: ” Probably the biggest benefit was the networking aspect, getting to
know people who work in the field, being able to talk to them in person. Another benefit was
finding out the interesting directions people are moving towards now, and getting lots of new
ideas for one’s own research.”
Brett Parker (joint with the Tropical Geometry program)
PhD: Stanford, 2005
Position prior to MSRI membership: UC Berkeley, visiting postdoc
Position after MSRI membership: postdoc at University of Zurich
Mentor: Yasha Eliashberg, Mark Gross, Michael Sullivan
Brett Parker received his Ph.D. from Stanford in 2005 under the supervision of Yakov Eliashberg.
His dissertation was titled ‘Holomorphic curves in Lagrangian torus fibrations’. In his
time at MSRI, Brett Parker worked on generalizing the symplectic sum formula for Gromov
Witten theory using the holomorphic curve theory of a new category called the category of
exploded manifolds. In this formalism, a symplectic manifold which is the result of a generalized
symplectic sum is is connected in a smooth family of exploded manifolds to an exploded
manifold in which the computation of Gromov Witten invariants reduces to a sum of relative
invariants over tropical curves. In the familiar case of a symplectic sum, the relative invariants
are Gromov Witten invariants relative to a symplectic submanifold, and the tropical curves
are in an interval and do not play an important role in understanding the symplectic sum
formula. In other cases however, the tropical curves are in a polytope of dimension as high
as half the dimension of the symplectic manifold, and sometimes the computation of Gromov
Witten invariants reduces to a combinatorial problem of a count of tropical curves. The relative
invariants involved can be regarded as a version of Gromov Witten invariants relative to
normal crossing divisors. At MSRI, Brett Parker was able to confirm a functorial connection
between his work in exploded manifolds and the work of Mark Gross and Bernd Siebert on
14