Our main result is that many triangles of Baumslag–Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag–Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length $\le \,3$. We obtain a lot of information about finite quotients, even when we cannot determine developability.

On établit quelques résultats généraux relatifs à la taille du groupe d’automorphismes de l’espace de Douady de points sur une surface, puis on étudie quelques propriétés des automorphismes provenant d’un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes.

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and, less completely, those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $\left( V,\,Q \right)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of Conf$\left( V,\,Q \right)$, the conformal group of the pair $\left( V,\,Q \right)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak{g}$, and furthermore a real form ${{\mathfrak{g}}_{\mathbb{R}}}$. The connected and simply connected Lie group ${{G}_{\mathbb{R}}}$ with $\text{Lie}\left( {{G}_{\mathbb{R}}} \right)\,=\,{{\mathfrak{g}}_{\mathbb{R}}}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood–Paley decomposition.

Let ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ be an increasing sequence of positive integers, and let $f\left( x \right)$ be a real function satisfying

1

$$f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty $$

If ${{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty $ the distribution of

2

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. In the case

$$1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty$$

there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$, and the limit distribution of (2).

In this paper we prove that any sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying $\lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$ contains a nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ such that for any function satisfying (1) the distribution of

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. This result is best possible: for any $\varepsilon >0$ there exists a sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying lim $\underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$ such that for every nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$.

Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in ${{\mathbb{R}}^{d}}$ with the potential ${{\left| x\,-\,y \right|}^{-s}},\,s\,\ge \,1$, where $\left| \cdot \right|$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for $s\,\ge \,2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N,\,N\to \,\infty $, where $L$ is the length of the curve.

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

We study projections in the corona algebra of $C\left( X \right)\,\otimes \,K$, where $K$ is the ${{C}^{*}}$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X\,=\,[0,\,1],\,[0,\,\infty ),\,(-\infty ,\,\infty ),\,\text{or}\,\text{ }\!\![\!\!\text{ 0,}\,\text{1 }\!\!]\!\!\text{ / }\!\!\{\!\!\text{ 0,}\,\text{1 }\!\!\}\!\!\text{ }$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in ${{K}_{0}}$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

Dans cet article, on énonce une variante du lemme fondamental pondéré d’Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l’aide de la méthode de descente, qui est conditionnelle : on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s’attend à ce que cette condition soit ultérieurement vérifiée.

Let $\Delta $ be a Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta $ are normal and Cohen–Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

In this article, we show to what extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.

In this paper we characterize the compactness of the commutator $\left[ b,\,T \right]$ for the singular integral operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$. More precisely, we prove that if $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$, the $\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$-closure of $C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$, then $\left[ b,\,T \right]$ is a compact operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for $1\,<\,p\,<\,\infty $ and $0\,<\,\lambda \,<\,n$. Conversely, if $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\left[ b,\,T \right]$ is a compact operator on the ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for some $p\,\left( 1\,<\,p\,<\,\infty \right)$, then $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $\left[ b,\,T \right]$ on ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

We consider the Randers space $({{V}^{n}},\,{{F}_{b}})$ obtained by perturbing the Euclidean metric by a translation, ${{F}_{b}}\,=\,\alpha \,+\,\beta $, where $\alpha $ is the Euclidean metric and $\beta $ is a 1-form with norm $b,\,0\,\le \,b\,<\,1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $({{V}^{3}},\,{{F}_{b}})$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b\,=\,0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $({{V}^{3}},\,{{F}_{b}})$ when $H\,=\,0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\,<\,b\,<\,\frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

A $\text{K}$-theoretic classification is given of the simple inductive limits of finite direct sums of the type I ${{C}^{*}}$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous ${{C}^{*}}$-algebras, each having spectrum a finite union of points and an open interval, and torsion ${{K}_{1}}$-group.)

We study the geometry of non-reductive four-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak $\text{KAM}$ theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti, and N. Yamada.