Previous Conferences & Workshops

We consider level sets of the Gaussian free field on the d-dimensional lattice, for d>2, above a given real-valued height h. This defines a percolation model with strong, algebraically decaying correlations. We prove a conjecture of Lebowitz...

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational 3-isogeny...

We consider level sets of the Gaussian free field on the $d$-dimensional lattice, for $d>2$, above a given real-valued height $h$. This defines a percolation model with strong, algebraically decaying correlations. We prove a conjecture of Lebowitz...

We start by showing that for any 1-parameter family of genus $g>2$ curves, the number of rational points is bounded by $g$, degree of the field, and the Mordell-Weil rank. Apart from the classical Faltings-Vojta-Bombieri method, the new input is a...

I will consider the asymptotic behavior of Laplacian eigenfunctions on a locally symmetric compact manifold as the volume approaches infinity. I will formulate a precise conjecture "à la Berry" and will describe some partial results obtained with...

Let M be a compact manifold with negative curvature. The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on M get equidistributed as the eigenvalue tends to infinity. A weaker version, called...

We explain the magic of the entire function $\Psi_p \in \mathbb{Z}[x] \cap \mathbb{Q}_p\{x\}$ defined by the functional equation \[x = \sum_{j=0}^{\infty} p^{-j}\Psi_p (p^{j} x)^{p^{j}}\] which satisfies $\Psi_p (\mathbb{Q}_p) \subset \mathbb{Z}_p$...