Erkao Bao, Definition of cylindrical contact homology
In this talk, I will give a definition of cylindrical
contact homology for 3-dimensional contact manifolds that admit non-degenerate
contact forms with no contractible Reeb orbits and show that it is
independent of choices. This is a joint work
(arXiv:1412.0276) with Ko Honda.
Olguta Buse, Symplectic packing stability in dimension four and above
In this talk we will discuss symplectic embeddings of ellipsoids or
unions of balls into arbitrary symplectic manifolds. We will establish
that sufficiently thin ellipsoids enjoy symplectic flexibility, that is,
they can be embedded into a target
symplectic manifold with a rational cohomology class as long as the volume
requirements are met. As a consequence we will generalize to all
dimensions Biran's four-dimensional result on symplectic packing
stability. For symplectic four manifolds with irrational
cohomology classes, we will show that packing stability by balls can be
established. This is joint with Richard Hind and Emmanuel Opshtein.
Dan Cristofaro-Gardiner, Higher dimensional symplectic embeddings and the Fibonacci staircase
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd-index Fibonacci numbers. I will discuss a version of this result that holds in all dimensions. This is joint work with Richard Hind.
Jean Gutt, Equivariant symplectic homology and periodic Reeb orbits
We will see how positive S1-equivariant symplectic homology gives information about the minimal number of periodic Reeb orbits on some contact manifolds. We first exhibit properties of positive S1-equivariant symplectic homology. New joint results with Kang on the standard sphere will be presented.
Kristen Hendricks, Invariance and computability of some spectral sequences from equivariant Floer homology
In the past few years, equivariant Floer cohomology has been used to construct many spectral sequences between Floer-type invariants of three-manifolds and knots. We will give an alternative formulation of equivariant Lagrangian Floer cohomology, which can be used to show several of these spectral sequences are invariants of their topological input data and/or explicitly computable. This is joint work in progress with R. Lipshitz and S. Sarkar.
Ailsa Keating, Homological Mirror Symmetry for singularities of type Tpqr
We present some homological mirror symmetry statements for the singularities of type Tpqr. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E. We will consider some symplectic invariants of the real four-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space P2, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different "flavours" of invariants (e.g., versions of the Fukaya category) match up on both sides.
Michael Khanevsky, Noncontinuity of surface quasimorphisms in the Hofer metric
There are several constructions of quasimorphisms on the Hamiltonian group of surfaces that were proposed by Gambaudo-Ghys, Polterovich and Py. These constructions are based on topological invariants either of individual orbits or of orbits of finite configurations of points and the quasimorphisms compute the average value of such invariants in the surface. We show that many quasimorphisms that arise this way are not Hofer continuous.
Tian-Jun Li, Space of surface configurations in uniruled symplectic 4-manifolds
Samuel Lisi, Symplectic Homology for Affine Algebraic Varieties
Symplectic homology is a version of Floer homology defined for symplectic manifolds with contact-type boundaries. This invariant detects a number of interesting properties of Weinstein domains, and is very rich in many examples; for instance, the symplectic homology of a cotangent bundle is isomorphic to the homology of the loop space (by Viterbo; Abbondandolo and Schwarz).
I will provide some background and motivation for studying symplectic homology, and will discuss the computation of symplectic homology for certain affine algebraic varieties in terms of relative Gromov-Witten invariants. This is joint work with Luis Diogo.
Cheuk-Yu Mak, Divisorial caps, uniruled caps and Calabi-Yau caps
We illustrate how a nice symplectic cap captures properties of symplectic fillings of a contact 3-manfiold. Three kinds of symplectic caps are introduced. Divisorial caps are motivated from compactifying divisors in algebraic geometry. Uniruled caps give strong restriction to symplectic fillings for a class of contact 3-manifolds strictly larger than the planar ones. Calabi-Yau caps, in particular, can be used to derive uniform Betti numbers bounds on Stein fillings of the standard unit cotangent bundle of any hyperbolic surface. This is a joint work with Tian-Jun Li and Kouichi Yasui.
Mark McLean, Log canonical threshold and Floer homology of the monodromy
The log canonical threshold of a hypersurface singularity is an important invariant which appears in many areas of algebraic geometry. For instance it is used in the minimal model program, has been used to prove vanishing theorems and it is related to the growth of solutions mod p^k, the Bernstein polynomial and also to the jet schemes. It can be calculated in many different ways. We show how to calculate the log canonical threshold using Floer homology of iterates of the monodromy map.
Jo Nelson, An integral lift of contact homology
Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH+ under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S1-equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.
Andres Pedroza, Hamiltonian loops and blow ups
We show how a Hamiltonian diffeomorphism can be lifted to the symplectic
one-point blow up. Also we show how a Hamiltonian loop on the base
manifold induces a nontrivial Hamiltonian loop on the blown up manifold.
Egor Shelukhin, Autonomous Hamiltonian flows and persistence modules
We discuss obstructions to including a Hamiltonian diffeomorphism into an autonomous flow that are robust in Hofer's metric. This is an application of symmetry and multiplicity in filtered Floer homology considered as a persistence module. Joint work with Leonid Polterovich.
Mohammad Tehrani, Normal crossing divisors in symplectic topology
In this talk, I will introduce topological notions of symplectic normal crossings divisor and configuration. These objects generalize the notion of normal crossings in algebraic geometry. I will introduce the notion of "regularization" which is a generalization of the "symplectic neighborhood theorem" for smooth symplectic divisors. Finally, I will talk about some applications such as generalizations of "symplectic sum and cut" constructions to the normal crossings case. (This is a joint work with Mark Mclean and Aleksey Zinger)
Weiwei Wu, Dehn twists exact sequences through Lagrangian cobordism
In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RPn and CPn, matching a mirror prediction due to Huybrechts and Thomas. This is a joint work with Cheuk-Yu Mak.