Joint IAS-PU Symplectic Geometry Seminar

Lagrangian Whitney sphere links

Ivan Smith
University of Cambridge
November 1, 2016
Let $n > 1$. Given two maps of an $n$-dimensional sphere into Euclidean $2n$-space with disjoint images, there is a $\mathbb Z/2$ valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on $n$, that this vanishes when the components are immersed Lagrangian spheres each with exactly one double point of high Maslov index. This is joint work with Tobias Ekholm.

Towards a theory of singular symplectic varieties

Aleksey Zinger
Stony Brook University
October 25, 2016
Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field. I will describe a new perspective on this question and motivate its appropriateness in the case of normal crossings singularities.

Length and width of Lagrangian cobordisms

Joshua Sabloff
Haverford College; Member, School of Mathematics
October 11, 2016
In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold.

Monotone Lagrangians in cotangent bundles

Luis Diogo
Columbia University
October 11, 2016
We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles. This is joint work with Mohammed Abouzaid.

Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

Kenji Fukaya
Stonybrook University
October 4, 2016
This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof.

Lagrangian cell complexes and Markov numbers

Jonny Evans
University College London
September 20, 2016
Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of ($2 \pi / p$). We call the resulting cell complex a '$p$-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels. In particular, we will see that a p-pinwheel admits a Lagrangian embedding in $CP^2$ if and only if $p$ is a Markov number.

A Heegaard Floer analog of algebraic torsion

Cagatay Kutluhan
University at Buffalo, The State University of New York; von Neumann Fellow, School of Mathematics
April 21, 2016
The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology. This is joint work with Grodana Matic, Jeremy Van Horn-Morris, and Andy Wand.

Symplectic embeddings and infinite staircases

Ana Rita Pires
Fordham University
April 15, 2016
McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid $E(2,3)$.

Classification results for two-dimensional Lagrangian tori

Georgios Dimitroglou-Rizell
University of Cambridge
April 7, 2016
We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone $S^2 \times S^2$.