Joint IAS-PU Symplectic Geometry Seminar

Infinite staircases and reflexive polygons

Ana Rita Pires
University of Edinburgh
July 3, 2020
A classic result, due to McDuff and Schlenk, asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. The work of McDuff and Schlenk has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase.

Distinguishing monotone Lagrangians via holomorphic annuli

Ailsa Keating
University of Cambridge
June 26, 2020
We present techniques for constructing families of compact, monotone (including exact) Lagrangians in certain affine varieties, starting with Brieskorn-Pham hypersurfaces. We will focus on dimensions 2 and 3. In particular, we'll explain how to set up well-defined counts of holomorphic annuli for a range of these families. Time allowing, we will give a number of applications.

Floer Cohomology and Arc Spaces

Mark McLean
Stony Brook University
June 12, 2020
Let f be a polynomial over the complex numbers with an isolated singular point at the origin and let d be a positive integer. To such a polynomial we can assign a variety called the dth contact locus of f. Morally, this corresponds to the space of d-jets of holomorphic disks in complex affine space whose boundary `wraps' around the singularity d times. We show that Floer cohomology of the dth power of the Milnor monodromy map is isomorphic to compactly supported cohomology of the dth contact locus.

Reeb orbits that force topological entropy

Abror Pirnapasov
Ruhr-Universität Bochum
June 5, 2020
A transverse link in a contact 3-manifold forces topological entropy if every Reeb flow possessing this link as a set of periodic orbits has positive topological entropy. We will explain how cylindrical contact homology on the complement of transverse links can be used to show that certain transverse links force topological entropy. As an application, we show that on every closed contact 3-manifold exists transverse knots that force topological entropy.

Real Lagrangian Tori in toric symplectic manifolds

Joé Brendel
University of Neuchâtel
June 5, 2020
In this talk we will be addressing the question whether a given Lagrangian torus in a toric monotone symplectic manifold can be realized as the fixed point set of an anti-symplectic involution (in which case it is called "real"). In the case of toric fibres, the answer depends on the geometry of the moment polytope of the ambient manifold. In the case of the Chekanov torus, the answer is always no. This can be proved using displacement energy and versal deformations.

Infinite staircases of symplectic embeddings of ellipsoids into Hirzebruch surfaces

Morgan Weiler
Rice University
June 5, 2020
Gromov nonsqueezing tells us that symplectic embeddings are governed by more complex obstructions than volume. In particular, in 2012, McDuff-Schlenk computed the embedding capacity function of the ball, whose value at a is the size of the smallest four-dimensional ball into which the ellipsoid E(1,a) symplectically embeds. They found that it contains an “infinite staircase” of piecewise-linear sections accumulating from below to the golden ratio to the fourth power. However, infinite staircases seem to be rare for more general targets.

Mirrors of curves and their Fukaya categories

Denis Auroux
Harvard University
May 22, 2020
Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a "stop" in its boundary at infinity.

Reflections on Cylindrical Contact Homology

Jo Nelson
Rice University
May 15, 2020
This talk beings with a light introduction, including some historical anecdotes to motivate the development of this Floer theoretic machinery for contact manifolds some 25 years ago. I will discuss joint work with Hutchings which constructs nonequivariant and a family Floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over Z and capture interesting torsion information.

The Geography of Immersed Lagrangian Fillings of Legendrian Submanifolds

Lisa Traynor
April 24, 2020
Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a “geography” question: What combinations of genus and double points can be realized by surfaces with boundary equal to K? I will discuss symplectic analogues of these questions: Given a Legendrian knot, what Lagrangian surfaces can it bound? What immersed Lagrangian surfaces can it bound?