May 25th: social gathering
June 1st: No meeting
June 8th: Joseph Breen [UCLA]: "An introduction to conformal symplectic geometry"
Do you lay awake at night, yearning for an alternate universe that has closed, exact symplectic manifolds? Do you feel intimidated by the wealth of Floer theory that permeates the subject, and long for a world in which no such thing exists (yet)? Look no further than conformal symplectic geometry! This talk is an introduction to conformal symplectic structures, a generalization of the familiar notion of symplectic structures. I will tell you about the differences and similarities with ordinary symplectic and contact geometry, and we will discuss some of the mysteries and unknowns in the field. The only prerequisite is a working knowledge of basic symplectic and contact geometry.
June 15th: Wenyuan Li [Northwestern U.]: "An introduction to Loose Legendrians in high dimensions"
Locally symplectic/contact manifolds all look the same, and thus they are very flexible, but globally do we still expect any flexibility? We will start by explaining what h-principles in the flexibility world are, and first state the h-principle for isotropic submanifolds (of subcritical dimension). Then we will explain what a loose Legendrian submanifold is, and why one should expect an h-principle for these Legendrian submanifolds. Finally, I will try to give a very biased overview on why we should care about loose Legendrians.
June 22nd: Alex Pieloch [Columbia U.]: "The growth rate of symplectic cohomology and affine varieties"
We discuss a result of McLean that states that any manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. The key ingredients include the symplectic invariance of the growth rate of symplectic cohomology and a description of the symplectic cochains of an affine variety in terms of a normal crossings compactification. First, we will give a cursory introduction to symplectic cohomology. After this, we will discuss the proof of McLean’s result, focusing primarily on McLean’s notion of symplectic normal crossings divisors and how appropriate symplectic neighborhoods of said divisors can give rise to explicit descriptions of the symplectic cochains of affine varieties.
June 29th: Mohamad Rabah [Stony Brook U.]: "Some Proofs of the Non-Squeezing Theorem"
This talk aims to give some flavors of the techniques and methods used in Symplectic Geometry. We start by introducing the necessary framework needed to give Gromov’s proof of the Non-Squeezing Theorem, which, by itself, can be used to define other invariants in Symplectic Topology (e.g. Gromov-Witten Classes). After that, we give a proof of the Non-Squeezing Theorem, as given by M.Gromov, using Minimal Surface Theory, followed by a second proof of the same theorem, but using Complex Geometry and without using Minimal Surface Theory. After that, we give a third proof where we use the notion of “Capacities”, but in disguise. If time permits, we give a fourth proof of the theorem, following Viterbo’s work.
July 6th: Leo Digiosia [ Rice U.]: "Cylindrical contact homology through examples"
In this talk we will describe the cylindrical contact homology associated to a closed three manifold. We will compute the cylindrical contact homology of the links of the simple singularities, a family of manifolds contactomorphic to quotients S^3/G, for finite subgroups G of SU(2). Throughout our computations we will note structural similarities between cylindrical contact homology and orbifold Morse homology when possible.
July 13th: Amanda Jenny, [University of Cambridge]: "Gromov-Witten axioms via implicit atlases"
We explain the concept of an implicit atlas and how they can be used to define a virtual fundamental class. Using this machinery, we define Gromov-Witten invariants of a closed symplectic manifold and show that these classes satisfy some of the axioms postulated by Kontsevich and Manin.
July 20th: Willi Kepplinger [U. of Vienna] "Metric Contact Topology and the Contact Sphere Theorem"
Riemannian Geometry and Topology (especially in low dimensions) have a long history of fruitful interactions in both directions. Nevertheless there are few results relating contact topology and the geometry/curvature properties of Riemannian metrics adapted to contact structures. One stunning result in this direction is the contact sphere theorem, a ''contact version'' of the famous topological/smooth sphere theorem in dimension 3, obtained by John Etnyre, Rafal Komendarczyk, and Patrick Massot. We aim to explain this result and give an idea of how contact topology and Riemannian geometry might interact in general.
July 27th: Shengzhen Ning [ [U. of Minnesota] "Gromov invariants on symplectic 4-manifolds without gauge theory"
The result of "Gr=SW" by Taubes had fruitful applications in studying symplectic 4-manifolds. In this talk we will describe a different approach towards computing the invariant "Gr" by Donaldson and Smith without using Seiberg-Witten theory. Given a Lefschetz fibration structure on the manifold, we will discuss another invariant "DS" by counting pseudoholomorphic sections of an associated space, which was proved to be equal to the invariant "Gr" by Michael Usher.
August 3rd: Mohan Swaminathan [Princeton U.] "Quantitative Gromov Compactness"
I will discuss a quantitative version of the Gromov compactness theorem in genus zero and some speculations on some possible applications of this result.
August 10th, 9am(PDT)=noon(EDT)=1am(Seoul):
BumJoon Sohn [Seoul National U.] "The Calabi invariant and the Reeb flows on S^3"
In this talk, we will introduce some results that show some interplay between area-preserving disk homeomorphism and Reeb-flows on tight 3-sphere. For dynamically convex contact forms on 3-sphere, there is a nicely embedded surface called a disk-like global surface of sections(GSS). For the GSS, the Poincare return map is an "Symplectic" map and Hofer-Wysocki-Zehnder used Franks' theorem to prove there exists two or infinitely many periodic Reeb orbits. We aim to introduce some results on Reeb flows using the Calabi-invariant(which is defined for an area-preserving map) for the special GSS and vice versa.
August 17th, 9am(PDT)=Noon(EDT)=6pm(Grenoble):
Bingyu Zhang [Institut Fourier, Grenoble] "What can we know from the conormal torus of a knot?"
A basic question is what can we know from the cotangent functor, which is a functor from differential topology to symplectic/contact topology. It is hard and far to be understood. The simplest case for the question is the case of knots in R^3. I will present how Shende uses microlocal sheaf theory to show the conormal torus is a complete knot invariant.
August 23rd, Monday, (1pm EDT)=(10 am PDT)=(6 pm Cambridge).
Nick Nho [Cambridge U.] "A gentle introduction to Hyperkahler geometry"
Hyperkahler geometry appears naturally throughout symplectic geometry. We will review basics of hyperkahler geometry, hyperkahler reduction and look at the main idea behind the construction of Hitchin moduli space.