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Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a…

Although the concept of defect of an analytic disc attached to a generic manifold of $\C^{n}$ seems to play a merely technical role, it turns out to be a rather deep and fruitful notion for the extendability of CR functions defined on the…

We prove that the additive group $(E^\ast,\tau_k(E))$ of an $\mathscr{L}_\infty$-Banach space $E$, with the topology $\tau_k(E)$ of uniform convergence on compact subsets of $E$, is topologically isomorphic to a subgroup of the unitary…

The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets.

We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$…

We prove that a Banach space $E$ has the compact range property (CRP) if and only if for any given $C^*$-algebra $\cal A$, every absolutely summing operator from $\cal A$ into $E$ is compact.

For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ such that the $x$-coordinates satisfy a linear equation. This classification enables us to establish a…

Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides…

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those…

For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert…

A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if $\Omega\subset\mathbb{R}^{n+1}$ is a convex body such that for every $n$-dimensional subspace $M\subset\mathbb{R}^{n+1}$ the…

We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and…

Motivated by the study of the large-scale geometry of topological groups, we investigate particular families of subsets of topological groups named group ideals. We compare different group ideals in the realm of locally compact groups. In…

We study finite subsets of $\ell_p$ and show that, up to nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains the spaces $\ell_p^n$, $n \in \mathbb{N}$.

Let E be a generic real submanifold of an almost complex manifold. The geometry of Bishop discs attached to E is studied in terms of the Levi form of E.

Local indices at isolated fixed points of a differentiable compact nonlinear map $T$ on Banach spaces will be discussed. These results are applied to establish the existence of nontrivial solutions. As an example, the existence of…

A closed subset $M$ of a Banach space $E$ is \ep, i.e., can be represented locally as the epigraph of a Lipschitz function, if and only if it is the level set of some locally Lipschitz function $f: E\to \R$, wich Clarke's generalized…

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

The special case of closed subsets of C^n is briefly discussed.

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed…