Joint IAS-PU Symplectic Geometry Seminar

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 $S^2 \times S^2$.

Homological Mirror Symmetry for singularities of type $T_{pqr}$

Ailsa Keating
Columbia University
March 10, 2016
We present some homological mirror symmetry statements for the singularities of type $T_{p,q,r}$. 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 $\mathbb P^2$, as suggested notably by Gross-Hacking-Keel.

Subflexible symplectic manifolds

Kyler Siegel
Stanford University
March 3, 2016
After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called "subflexible", which are not flexible but become so after attaching some Weinstein handles. For example, the standard symplectic ball has a Weinstein subdomain with nontrivial symplectic topology. These are exotic symplectic manifolds with vanishing symplectic cohomology.

Positive loops---on a question by Eliashberg-Polterovich and a contact systolic inequality

Peter Albers
Universität Münster
February 25, 2016
In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether $C^0$-small positive loops exist. We give a negative answer to this question. Moreover we give sharp lower bounds for the size which, in turn, gives rise to a $L^\infty$-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no $L^2$-contact systolic inequality exists.

Floer theory revisited

Mohammed Abouzaid
Columbia University
February 4, 2016
I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I will explain the consequences of quantum cohomology, and the expected applications of this theory.