We consider Willmore surfaces in ${\mathbb{R}}^3$ with an isolated singularity of finite density at the origin. We show that locally, the surface is a union of finitely many multivalued graphs, each with a unique tangent plane at zero and with second fundamental form satisfying ${\displaystyle |A(x)|\le C_{\epsilon }|x{|}^{-1+1/{\theta }_0-\epsilon },\forall \epsilon >0,}$ where ${\theta }_0\in \mathbb{N}$ is the maximal multiplicity. Examples of branched minimal surfaces show that this estimate is optimal up to the error $\epsilon >0$

We classify joinings between a fairly general class of higher-rank diagonalizable actions on locally homogeneous spaces. In particular, we classify joinings of the action of a maximal $\mathbb{R}$-split torus on $G/\Gamma$ with $G$ a simple Lie group of $\mathbb{R}$-rank at least $2$ and $\Gamma <G$ a lattice. We deduce from this a classification of measurable factors of such actions as well as certain equidistribution properties

We use the recent proof of Jacquet's conjecture due to Harris and Kudla [HK] and the Burger-Sarnak principle (see [BS]) to give a proof of the relationship between the existence of trilinear forms on representations of ${\mathrm{GL}}_2(k_u)$ for a non-Archimedean local field $k_u$ and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in [P1, Theorem 1.4]. The method used is very flexible and gives a global proof of a theorem of Saito and Tunnell about characters of ${\mathrm{GL}}_2$ using a theorem of Waldspurger [W, Theorem 2] about period integrals for ${\mathrm{GL}}_2$ and also an extension of the theorem of Saito and Tunnell by this author in [P3, Theorem 1.2] which was earlier proved only in odd residue characteristic. In the appendix to this article, H. Saito gives a local proof of Lemma 4 which plays an important role in the article

In this article, we consider the problem of finding upper bounds on the minimum norm of representatives in residue classes in quotient $O/I$, where $I$ is an integral ideal in the maximal order $O$ of a number field $K$. In particular, we answer affirmatively a question of Konyagin and Shparlinski [KS], stating that an upper bound $o(N(I))$ holds for most ideals $I$, denoting $N(I)$ the norm of $I$. More precise statements are obtained, especially when $I$ is prime. We use the method of exponential sums over multiplicative groups, essentially exploiting some new bounds obtained by the authors

We obtain global well-posedness, scattering, and global $L_{t,x}^{2(n+2)/(n-2)}$ space-time bounds for energy-space solutions to the energy-critical nonlinear Schrödinger (NLS) equation in ${\mathbb{R}}_t\times {\mathbb{R}}_x^n$, $n\ge 5$