Let M be a compact symplectic manifold endowed with a Hamiltonian torus action. I will discuss how to extend the well-known result that components of a moment map for the action are Morse-Bott functions on M to make arguments about the topology of a partial level set. This may be used to make conclusions about orbifold invariants of symplectic quotients. The talk will include many examples, including toric orbifolds and quotients of coadjoint orbits by subtori of the maximal torus.

I will discuss an algebraic relationship between the Khovanov homology of certain tangles in product sutured manifolds and the Heegaard Floer homology of their sutured double-branched covers. This relationship implies that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.

Furthermore, certain TQFT operations (cutting, stacking) on the tangles correspond naturally to geometric operations (generalized plumbing) on the sutured double-branched covers, and the algebraic connection between Khovanov and Heegaard Floer homology behaves well with respect to these operations.

This is joint work with Stephan Wehrli.

For each n>6, we construct (mainly using handle attaching) a list of contact manifolds C1,C2,C3,... diffeomorphic to the sphere of dimension 2n-1 such that there is no algorithm that tells you which of these manifolds are contactomorphic to C1.

The idea of the proof is to reduce this problem to the word problem for groups using an invariant called the growth rate of Symplectic homology.

Fixed-point Floer homology is an invariant of symplectomorphisms. Its rank is a lower bound for the number of fixed points of non-degenerate symplectomorphisms Hamiltonian-isotopic to the original. Seidel conjectured that fixed point Floer homology of the monodromy of a symplectic Lefschetz fibration over the disc is isomorphic to the homology of a Hochschild-type complex, built from the directed A-infinity category of vanishing cycles and the Morse complex of the fibre. We discuss recent progress on this conjecture, and propose a connection to Ozsvath-Szabo's link-surgeries spectral sequence in Heegaard Floer homology.

I will show some examples of calculating monotone Floer homology from a general strip shrinking isomorphism in quilted Floer homology (for sequences of Lagrangian correspondences). Examples include the Clifford torus in CP^n (previously known by Cho) and nondisplaceable T^{n-k}\times S^{2k-1} in CP^n\times CP^{k-1}. Moreover, the bijection of trajectory moduli spaces can be somewhat generalized to multiply covered compositions of correspondences, yielding e.g. calculations of the Floer homology between Clifford tori and RP^n in CP^n (confirming work by Allston). Finally, we can prove Hamiltonian nondisplaceability of the Chekanov/Polterovich torus in S^2\times S^2; using symmetries and twisted coefficients.

We describe work-in-progress which aims to study the action of the mapping class group of a surface on the moduli space of twisted SU(2)-representations of its fundamental group (equivalently the moduli space of odd determinant rank two stable bundles). The approach involves Fukaya categories of quadric hypersurfaces in projective space, and motivation from homological mirror symmetry.

Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to SL(n) and tensor products of the standard and dual representations. I will place their geometric setups in a natural, general framework using the Beilinson-Drinfeld Grassmannian and the geometric Satake correspondence.

I'll describe new bigraded invariants of a framed link in a 3-manifold, based on a generalization of the surgery exact triangle in monopole Floer homology. The construction relates the topology of link surgeries to the combinatorics of polytopes called graph associahedra. For a link in the 3-sphere, we obtain a sequence of vector spaces, interpolating between versions of Khovanov homology and monopole Floer homology of the branched double cover. This perspective also yields a simple, topological proof that odd Khovanov homology is mutation invariant.

The moduli spaces of curves with level structures are the finite covers of the moduli spaces of curves associated with the linear congruence subgroups of the mapping class group. We will describe a sequence of results about the low-dimensional cohomology groups of these linear congruence subgroups which yield a complete description of the complex line bundles on the corresponding moduli spaces.