Program 2015

Lecture Series

Hamiltonian diffeomorphisms and Floer theory

Michael Usher

Lecture 1

Abstract: This will be an introductory lecture on Hamiltonian diffeomorphisms and their Floer homology. I will recall some basic notions of symplectic geometry and the Hamiltonian diffeomorphism group, and explain how Floer homology was developed to prove a variant of Arnold’s conjecture on symplectic fixed points.

Lecture 2.

Abstract: A remarkable feature of the Hamiltonian diffeomorphism group of a symplectic manifold is a bi-invariant Finsler-type metric discovered by Hofer in 1990. In this lecture I will introduce the metric and discuss ways in which Floer theory can be used to obtain lower bounds for it.

Lecture 3.

Abstract: Hofer’s metric on the Hamiltonian diffeomorphism group naturally induces a pseudometric on the orbit of any closed subset under the group. I will explain how the behavior (for instance, degeneracy or nondegeneracy) of this metric is related to other geometric properties of the subset.

Lecture 4.

Abstract: It has recently been observed that the language of persistent homology is well-suited to studying the filtration on the Hamiltonian Floer complex, at least in the symplectically aspherical case. I will discuss some joint work with Jun Zhang which extends basic notions of persistent homology to the context of chain complexes over Novikov rings (such as those arising in Floer homology for non-symplectically-aspherical manifolds), and mention some simple symplectic corollaries of the main structural results.

Invited lectures

Perfect Reeb flows and action index relations

Başak Gürel

Abstract: In this talk we will discuss a recent work where we study non-degenerate Reeb flows arising from perfect contact forms, i.e., the forms with vanishing contact homology differential. The main results are upper bounds on the number of simple closed Reeb orbits for such forms on a variety of contact manifolds and certain action-index resonance relations for the standard contact sphere.

Symplectic Embeddings of products

Richard Hind

Abstract: In their 2012 paper McDuff and Schlenk completely solved the existence problem for symplectic embeddings of 4-dimensional ellipsoids into balls. In other words, they calculated the function

$c(x) = \inf \{R | E(1,x) \hookrightarrow B^4(R) \}.$

Here an ellipsoid inside the standard symplectic Euclidean space is written as $E(a,b) = \{ \frac{\pi}{a}(p_1^2 + q_1^2) + \frac{\pi}{b}(p_2^2+q_2^2) < 1\}$ and $B^4(R)=E(R,R)$ is a ball.

For a fixed $n \ge 3$ we can define the function

$f(x)= \inf \{R | E(1,x) \times \mathbb R^{2(n-2)} \hookrightarrow B^4(R) \times \mathbb R^{2(n-2)} \}.$

I will talk about some constructions and obstructions which give upper and lower bounds respectively for $f(x)$ . It is clear that $f(x) \le c(x)$ but it turns out we have equality precisely when $x \le \tau^4$ , the fourth power of the golden ratio. This is work in progress with Daniel Cristofaro-Gardiner.

TBA

Yasha Savelyev

Abstract: TBA

Contributed Lectures

Orthosymplectic and symplectogonal supergeometries

Óscar Guajardo

Abstract: In this talk we’ll introduce even and odd superriemannian and
supersymplectic structures on smooth supermanifolds, comparing them to the
classical cases. The approach is closer to classical differential geometry in
that it deals with bundles and connections rather than sheaves, and it is based
on the author’s Master’s thesis which can be found at http://arxiv.org/abs/14010.7857.

Moment-angle manifolds and higher dimensional contact manifolds

Yadira Lizeth Barreto Felipe

Abstract: We construct new examples of contact manifolds in arbitrarily large
dimensions. This manifolds are related to the classical moment-angle manifolds.

Ricci solitons on symplectic and contact manifolds

Jonatán Torres Orozco

Abstract: In 1983 Richard Hamilton introduced the Ricci Flow as an approach to
prove Thurston’s Geometrization and Poincaré Conjectures in dimension 3.
It is expected that Ricci Flow helps to understand higher dimension
geometrization problems. Einstein metrics are fixed points for the Ricci flow,
since fixed points must have constant average scalar and Ricci curvatures. In order
to understand singularities to the Ricci flow research is strongly related to its
fixed points and a generalization, Ricci Solitons. They often arise as dilation
limits of singularities in the flow.

Poisson structures in smooth4-manifolds

Pablo Suárez Serrato

Abstract: I will explain a construction of Poisson structures on smooth 4-manifolds
associated to underlying broken Lefschetz fibrations. We will provide explicit local
formulae for the Poisson bivector. Some questions regarding these structures will be presented.
This is joint work with L. García-Naranjo and R. Vera.

Hofer’s metric on the space of curves

Michael Khanevsky

Abstract: Given a curve in a surface, we consider all curves Hamiltonian isotopic to the given one. This space admits a metric induced by Hofer’s metric on the Hamiltonian group. We present several examples and discuss tools that can be used to study geometry of such spaces.