Shellability of noncrossing partition lattices

We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three.

Valuations, intersections et fonctions de Belyi

We present in this article several possibilities to approach the height of an algebraic curve defined over a number field : as an intersection number via the Arakelov theory, as a limit point of the heights of its algebraic points and, finally, using the minimal degree of Belyi functions.

Petits points et conjecture de Bogomolov

This paper is devoted to the statement known as the Bogomolov conjecture on small points. We present the outline of Zhang’s proof of the generalized version of the conjecture. An explicit bound for the height of a non-torsion variety of an abelian variety is obtained in the frame of Arakelov theory. Some further developments are mentioned.

Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This ‘growth alternative’ conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied.

Polynomial and linearized normal forms for almost periodic differential systems

"Vegeu el resum a l'inici del document del fitxer adjunt."

On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift

We consider the Kudla-Millson lift from elliptic modular forms of weight (p+q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p,q). We study the L²-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L²-norm of the lift, which often implies that the lift is injective. For O(p,2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift.

Stable rank of leavitt path algebra

We characterize the values of the stable rank for Leavitt path algebras, by giving concrete criteria in terms of properties of the underlying graph.

The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-Algebras

We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C¤-algebras. In particular, our results apply to the largest class of simple C¤-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Z-stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C¤-algebras. We also prove in passing that the Kuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for all simple unital C¤-algebras of interest.

The bogomolov conjecture for totally degenerate Abelian varieties

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place.

Tropical varieties for non-archimedean analytic spaces

"Vegeu el resum a l'inici del document del fitxer adjunt."

Shelah's singular compactness theorem

We present Shelah’s famous theorem in a version for modules, together with a self-contained proof and some examples. This exposition is based on lectures given at CRM in October 2006.

┴N as an Abstract Elementary Class

"Vegeu el resum a l'inici del documents del fitxer adjunt"

Minimal lenght elements of Thompson's groups F(p)

We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.