We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, and a heat equation with dynamic boundary conditions.

Soit $c\>1$, $p$ un nombre premier et $𝒜$ une partie de $\mathbb{Z}/p\mathbb{Z}$ de cardinal supérieur à $c\sqrt{p}$ telle que pour tout sous-ensemble non vide $ℬ$ de $𝒜$, on a ${\sum }_{b\in ℬ}b\ne 0$. On montre qu’il existe $s$ premier à $p$ tel que l’ensemble $s.𝒜$ est très concentré autour de l’origine et qu’il est presque entièrement composé d’éléments de partie fractionnaire positive. Plus précisément, on a$$\sum _{a\in 𝒜}∥\frac{sa}{p}∥\<1+O(p^{-1/4}lnp)\text{et}\sum _{\begin{array}{c}a\in 𝒜,\\ \{sa/p\}\ge 1/2\end{array}}∥\frac{sa}{p}∥=O(p^{-1/4}lnp).$$On montre également que les termes d’erreurs ne peuvent être remplacés par $o\left(p^{-1/2}\right)$.

We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.

Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1,1)$. The quotient ${ℍ}^{n+1}/\Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

During the last decade, several quite unexpected applications of the Schmidt Subspace Theorem were found. We survey some of these, with a special emphasize on the consequences of quantitative statements of this theorem, in particular regarding transcendence questions.

We develop the basic theory of the Weyl symbolic calculus of pseudodifferential operators over the $p$-adic numbers. We apply this theory to the study of elliptic operators over the $p$-adic numbers and determine their asymptotic spectral behavior.

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PS{L}_2\left(\mathbb{Z}\right)\setminus PS{L}_2\left(ℝ\right)$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PS{L}_2\left(\mathbb{Z}\right)\setminus ℍ$. The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S{L}_2\left(ℝ\right)$. In the proof the key estimates come from applying...