In this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the “finite dimensional” case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy–Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

Recently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb{R}\times {{S}^{n-1}}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${{\mathbb{C}}^{n}}$ is foliated by round $\left( n-1 \right)$-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${{\mathbb{C}}^{n}}$. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

Nous démontrons que tous les plongements d’une variété compacte sans bord et simplement connexe de dimension quatre dans la sphère de dimension six sont concordants.

It is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products.

We provide concise series representations for various $\text{L}$-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying $\text{L}$-series.

We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma _{0}^{+}\left( p \right)\,\,:={{\Gamma }_{0}}\left( p \right)+{{w}_{p}}$, where $p$ is a prime and ${{w}_{p}}$ is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on ${{\Gamma }_{1}}\left( p \right)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

We show that there exists a closed infinite dimensional subspace of ${{H}^{\infty }}\left( {{B}^{n}} \right)$ such that every function of norm one is universal for some sequence of automorphisms of ${{B}^{n}}$.

We construct several examples of higher-dimensional Calabi–Yau manifolds and prove their modularity.

Let $v\,\ge \,k\,\ge \,1$ and $\lambda \,\ge \,0$ be integers. A block design$\text{BD}\left( v,\,k,\,\lambda \right)$ is a collection $\mathcal{A}$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda $ elements of $\mathcal{A}$. More generally, for a fixed simple graph $G$, a graph design$\text{GD}\left( v,\,G,\,\lambda \right)$ is a collection $\mathcal{A}$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda $ elements of $\mathcal{A}$. A famous result of Wilson says that for a fixed $ $ and $\lambda $, there exists a $\text{GD}\left( v,\,G,\,\lambda \right)$ for all sufficiently large $ $ satisfying certain necessary conditions. A block (graph) design as above is resolvable if $\mathcal{A}$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\text{BD}\left( v,\,k,\,\lambda \right)$, yet for over twenty years the analogous problem for resolvable $\text{GD}\left( v,\,G,\,\lambda \right)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

We classify the left-invariant metrics with nonnegative sectional curvature on $\text{SO}\left( 3 \right)$ and $U\left( 2 \right)$.

A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained.

Consider a real potential $V$ on ${{\text{R}}^{d}}$, $d\ge 2$, and the Schrödinger equation:

$$\left( \text{LS} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i{{\partial }_{t}}u+\Delta u-Vu=0,{{u}_{\upharpoonright }}_{t=0}={{u}_{0}}\in {{L}^{2}}$$

In this paper, we investigate the minimal local regularity of $V$ needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of $1/2$ derivative) on solutions of $\left( \text{LS} \right)$. Prior works show some dispersive properties when $V$ (small at infinity) is in ${{L}^{d/2}}$ or in spaces just a little larger but with a smallness condition on $V$ (or at least on its negative part).

In this work, we prove the critical character of these results by constructing a positive potential $V$ which has compact support, bounded outside 0 and of the order ${{\left( \log \left| x \right| \right)}^{2}}/{{\left| x \right|}^{2}}$ near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator $P:=-\Delta +V$.

The elementary construction of $V$ consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator $P$ admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.

We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$, say $d$, one of these two polynomials is irreducible and another has an irreducible factor of degree $d$, so that $\alpha =M\left( P \right)-bM\left( Q \right)$ with irreducible polynomials $P,Q\in \mathbb{Z}\left[ X \right]$ of degree $d$ and a positive integer $b$. Finally, if $d\le 3$, then one can take $b=1$.

It is shown that Schatten $p$-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative ${{L}_{p}}$-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative ${{L}_{p}}$ spaces. As a consequence, the class of non-commutative ${{L}_{p}}$-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative ${{L}_{p}}$-spaces. For $p\,=\,1$ this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Given an odd surjective Galois representation $\varrho :{{\text{G}}_{\mathbb{Q}}}\to \text{PG}{{\text{L}}_{2}}\left( {{\mathbb{F}}_{3}} \right)$ and a positive integer $N$, there exists a twisted modular curve $X{{\left( N,3 \right)}_{\varrho }}$ defined over $\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves of degree $N$ realizing $\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$.

In this paper, we study the tensor product $\pi ={{\sigma }^{\min }}\otimes {{\sigma }^{\min }}$ of the minimal representation ${{\sigma }^{\min }}$ of $O\left( p,q \right)$ with itself, and decompose $\pi$ into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\mathbb{Z}$-module ${{\mathcal{D}}_{A}}\left( W \right)$ of the group algebra $\mathbb{Z}W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci–Reutenauer algebra.

Let $M$ be a symplectic 4-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.