Symplectic geometry has one of its origins in Hamiltonian dynamics. In the late 60s Arnold made a fundamental conjecture about the minimal number of periodic orbits of Hamiltonian vector fields. This is a far-reaching generalization of Poincaré's last geometric theorem and completely changed the field of symplectic geometry. In the last 30 years symplectic geometry has been tremendously developed due to the theory of holomorphic curves by Gromov and Floer homology theory by Floer. I will give a gentle introduction to the field of symplectic geometry and explain how modern methods give rise to existence results for periodic orbits and discover rigidity phenomena in symplectic geometry.

Given an action of a finite group G on an affine space A^n, the quotient variety A^n/G have been a subject of interest for over a century, partially motivated by Noether's problem concerning its rationality. Weaker properties of this variety were extensively studied, relating it to the Inverse Galois problem, parametrization of Galois extensions and essential dimension. One of its weakest possible properties, (very) weak approximation, is the subject of a more recent conjecture by Colliot-Th'el`ene. We shall discuss the above rationality properties, the conjecture, and the (recent) progress towards it. 

Spatial sampling is particularly important for environmental statistics. A simple reasoning developed in Grafström and Tillé (2013) shows that under a self-correlated linear model, it is more efficient to select a well-spread sample in space. If we select two neighbouring units in a sample, we will tend to collect partially redundant information. Grafström and Lundström (2013) discuss at length the concept of spreading, also known as spatial balancing and its implication on estimation.  The Generalized Random Tesselation Sampling GRTS design has been proposed by Stevens Jr. and Olsen (1999, 2004, 2003) to select spread samples. The pivotal method has been proposed by Deville and Tillé (2000). Grafström et al. (2012) proposed to use the pivotal method for spatial sampling. This method, called the local pivotal method, consists, at each step, in comparing two neighboring units. If the probability of one of these two units is increased, the probability of the other is decreased, which induces a repulsion between the units. The natural extension of this idea is to confront a group of units. The local pivot method was generalized by Grafström and Tillé (2013) to provide samples that are both well-balanced in space and balanced on the totals of the auxiliary variables. This method is called the local cube method. We also propose a new method that enables us to select spreader samples that all existing methods and allows the construction of periodic sampling plans when these plans exist.

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number K of clusters of sizes linearly growing with the global population of items n. Every pair of items is labeled independently at random, and label ℓ appears with probability p(i,j,ℓ) between two items in clusters indexed by i and j, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most smisclassified items in average under the general LSBM and for any s=o(n), which solves one open problem raised in Abbe et al 2015. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within O(n polylog(n)) computations and without the a-priori knowledge of the model parameters.