Imagine it is Autumn in the forest, and randomly shaped leaves fall sequentially at random onto the ground until it is completely covered. The visible parts of leaves on the ground at a given instant then form a random tessellation of the plane. Mathematically, the leaves and their times of arrival are modelled as an independently marked Poisson process in space-time with the marks determining their shapes. This dead leaves (or confetti) model was proposed by Matheron in 1968 and has applications in modelling natural images and in materials science; see [1]. The one-dimensional version of the model (leaves on the line) can be obtained by simply taking a linear section through the two-dimensional tessellation but is also open to other interpretations, for example the tree-trunks visible from the edge of the forest. We discuss new and old results on some or all of the following in both one and two dimensions: * Exact formula for the number of cells of the tessellation per unit volume. * Asymptotic variance, CLT and time evolution for the total number of cells visible in a large window. * Analogous results in one dimension. * In two dimensions, similar results for the total length of cell boundaries within a window. * In one dimension, the distribution of the size of a `typical' cell. We make heavy use of the Mecke formula from the theory of Poisson processes; see e.g. [3]. Also relevant is a variant of the classical `Buffon's needle' problem. We also build on earlier work on items 1 and 3 by Cowan and Tsang [2].

The canonical white noise W on a compact manifold M can be written as a random sum over the Laplace eigenfunctions on M with iid standard normal coefficients. By truncating this sum, one obtains a smooth Gaussian field on M: the cut-off white noise. In the past decade, much has been said about the geometry and topology of this random field. In particular, its average statistics are in a sense universal. On the other hand, one can generalize the above definition and study fields defined as sums over Laplace eigenfunctions with independent normal coefficients whose variance decays at a certain rate. These are known as fractional gaussian fields. This wide family contains for instance, the periodic Brownian motion and the Gaussian Free Field. We will discuss the geometry of the level set of these fields.

This talk focuses on a multidimensional SDE where the drift is an element of a fractional Sobolev space with negative order, hence a distribution. This SDE admits a unique weak solution in a suitable sense - this was proven in [Flandoli, Issoglio, Russo (2017)]. The aim here is to construct a numerical scheme to approximate this solution. One of the key problems is that the drift cannot be evaluated pointwise, hence we approximate it with suitable functions using Haar wavelets, and then apply (an extended version of) Euler-Maruyama scheme. We then show that the algorithm converges in law, and in the special 1-dimensional case we also get a rate of convergence (and in fact convergence in L^1). This talk is based on a joint work with T. De Angelis and M. Germain.