Calabi quasimorphisms

In this talk, we start by introducing the general concept of quasimorphism which is an approximated version of homomorphism. It follows by an interesting example of group in symplectic geometry - Ham (M, \omega), the group of Hamiltonian diffeomorphism group. We study some algebra and analysis aspects of this group. Based on these properties, we can associate a quasimorphism to this group which is called Calabi quasimorphism. In general, construction and computation of this quasimorphism is difficult, but in the case of the symplectic sphere, we can give an explicit formula of the Calabi quasimorphism on the sphere in terms of the Reeb graph. Moreover, this quasimorphism will lead to the construction of symplectic quasi-state which plays an important role in the development of modern symplectic geometry according to the pioneer work from Entov and Polterovich. Details of proof will be given and also some open questions will be mentioned. Notes [E-P]

Geometric Quantization

The aim of this talks is to explain some of the motivations behind geometric quantization of Kähler manifolds. After having defined what is meant by quantization and explained the case of a cotangent bundle, Yohann shows how trying to quantize a general symplectic manifold from the most naive approach and successively correcting the construction naturally leads to the introduction of the relevant structures (prequantum bundle, Kähler structure). Notes [W]

Partial symplectic quasistates and their applications to non-displaceability

We speak on the theory of partial symplectic quasi-states constituted by Entov-Polterovich's papers. Precisely we speak on definition, construction and fundamental properties of spectral invariants, construction of partial quasistates by spectral invariants, definition and properties of heavy and superheavy subsets, the idea of stem which is one of the methods to find superheavy subsets and some examples where we prove a non-displeceability result by using partial symplectic quasi-states. The above examples contain the Clifford torus in the complex projective space, that is non displaceable by symplectomorphisms. We also show that the product of a median curve and longitude curve in a 2-torus times the Clifford torus is non displeceable by symplectomorphisms, this is part of Morimichi's results about the subject. [E-R] Notes

Lagrangian Floer Theory [A]

The Quantum Euler Class and the Quantum cohomology of the Grassmannians

In this talk we define what a Frobenius algebra and its characteristic element mean. We explain the following result: a finite dimensional Frobenius algebra is semisimple if and only if its characteristic element is a unit. Then we apply this result to the case of the quantum cohomology of the complex Grassmamnnians and we show why its specializations are semisimple Frobenius algebras. Notes [Ab]

Delta-non degeneracy and submanifolds

Nowadays, symplectic geometry very often studies the subset of a symplectic manifold by classified as displaceable or non-displaceable. But in some case, it is hard to get more information (for instance, symplectic quasistates is always zero on displaceable subset). In this talk, we will introduce another way to study subsets of symplectic manifolds motivated by the question of non-degeneracy of Lagrangian-Hofer metric (later developed by Chekanov, so called CH-metric). We define weightless and CH-rigid subset and give testing properties for both of them. Moreover, we will see from these properties, a large class of submanifolds as supporting examples can be proved to be weightless or CH-rigid. This is part of work done by Usher. Notes [U1], [U2]

Conjugation-invariant norms and symplectic contact rigidity

Burgaro-Ivanov-Polterovich induced the notion of conjugation-invariant norm on group. Commutator length and Hofer's norm are famous examples of conjugation invariant norms. On the ther hand, some mysterious phenomena such as non-squeezing, non-displaceability in symplectic topology are often called "symplectic rigidity." The speaker will introduce relationships between conjugation-invariant norms and symplectic contact rigidity via some interesting result. If we have sufficient time, he introduce his original result related to conjugation-invariant norms. [B-I-P], [F-P-R]. Notes

Geometric quantization of compact Kahler Manifolds

In the first talk I explained why, when the phase space is a compact Kahler manifold, it is reasonable to construct the Hilbert spaces of quantum mechanics as spaces of holomorphic sections of powers of some complex line bundle endowed with a connection of curvature -i times the symplectic form. The purpose of the present talk is to detail this construction and work out a few concrete examples. [H], [M],[W]. Notes

Enumerative geometry, symplectic geometry and Welshinger invariants

The goal of this talk is to present Welshinger invariants as described in the paper "Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry" by Jean-Yves-Welshinger. We start off with a brief introduction to classical enumerative geometry and then continue by describing certain enumerative invariants in symplectic topology. The final part of the talk is dedicated to a description of Welschinger invariants and sketch of proof of their invariance. [We] Notes

A short introduction to Schubert Calculus.

In this talk we motivate the use of Schubert calculus to solve some enumerative problems such as the "four line problem". We define Schubert cells and Schubert varieties in a Grassmannian manifold and we define the corresponding Schubert classes in its cohomology ring. We explain the relation between the cohomology ring of the Grassmannian with the ring of symmetric functions, and we define Schur functions. We show have these relations are used to solve the four line problem. [F] Notes

Contact Homology

Notes