Noncommutative geometry on trees and buildings

We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum

Dynamical Systems on Spectral Metric Spaces

Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak*-topology on the state space of A. A *-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra AxZ, characterizing the metric properties of the dynamical system ? If $\alpha$ is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,$\alpha$) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this new system are discussed. Applications, in number theory, in coding theory are given at the end

Gitter und Anwendungen

The meeting focussed on lattices and their applications in mathematics and information technology. The research interests of the participants varied from engineering sciences, algebraic and analytic number theory, coding theory, algebraic geometry to name only a few

Intersection matrices for the minimal regular model of X0(N){X}_0(N) and applications to the Arakelov canonical sheaf

Let $N>1$ be an integer coprime to $6$ such that $N\notin\{5,7,13\}$ and let $g=g(N)$ be the genus of the modular curve $X_0(N)$. We compute the intersection matrices relative to special fibres of the minimal regular model of $X_0(N)$. Moreover we prove that the self-intersection of the Arakelov canonical sheaf of $X_0(N)$ is asymptotic to $3g\log N$, for $N\to+\infty$.Comment: 27 pages. The main results have been improved for any N coprime to 6. Moreover there is an Appendix with the drawings of the special fibre

Arithmetical problems in number fields, abelian varieties and modular forms

La teoria de nombres, una àrea de la matemàtica fascinant i de les més antigues, ha experimentat un progrés espectacular durant els darrers anys. El desenvolupament d'una base teòrica profunda i la implementació d'algoritmes han conduït a noves interrelacions matemàtiques interessants que han fet palesos teoremes importants en aquesta àrea. Aquest informe resumeix les contribucions a la teoria de nombres dutes a terme per les persones del Seminari de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus resultats són citats en connexió amb l'estat actual d'alguns problemes aritmètics, de manera que aquesta monografia cerca proporcionar al públic lector una ullada sobre algunes línies específiques de la recerca matemàtica actual.Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research

Ganzzahlige quadratische Formen und Gitter

[no abstract available

Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS_3/CFT_2 correspondence

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete—the Bruhat–Tits tree for PGL(2,Qp)—but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat–Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu–Takayanagi formula for the entanglement entropy appears naturally