Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants

We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories. We introduce virtual fundamental classes and thus obtain weighted Gromov-Witten invariants. We show that by including gravitational descendants, one obtains an \LL-algebra as introduced in [LM04] as a generalization of a cohomological field theory.Comment: 28 pages; v2: references added and updated, addressed referee comments; to appear in Moscow Math Journa

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

The Riemann hypothesis: The great pending mathematical challenge

The Riemann hypothesis is an unproven statement referring to the zeros of the Riemann zeta function. Bernhard Riemann calculated the first six non-trivial zeros of the function and observed that they were all on the same straight line. In a report published in 1859, Riemann stated that this might very well be a general fact. The Riemann hypothesis claims that all non-trivial zeros of the zeta function are on the the line x = 1/2. The more than ten billion zeroes calculated to date, all of them lying on the critical line, coincide with Riemann's suspicion, but no one has yet been able to prove that the zeta function does not have non-trivial zeroes outside of this line

Implementation issues in product line scoping

Often product line engineering is treated similar to the waterfall model in traditional software engineering, i.e., the different phases (scoping, analysis, architecting, implementation) are treated as if they could be clearly separated and would follow each other in an ordered fashion. However, in practice strong interactions between the individual phases become apparent. In particular, how implementation is done has a strong impact on economic aspects of the project and thus how to adequately plan it. Hence, assessing these relationships adequately in the beginning has a strong impact on performing a product line project right. In this paper we present a framework that helps in exactly this task. It captures on an abstract level the relationships between scoping information and implementation aspects and thus allows to provide rough guidance on implementation aspects of the project. We will also discuss the application of our framework to a specific industrial project

Uniformization of triangle modular curves

In the present article, we determine explicit uniformizations of modular curves attached to triangle Fuchsian groups with cusps. Their Hauptmoduln are obtained by integration of non-linear differential equations of the third order. Series expansions involving integral coefficients are calculated around the cusps as well as around the elliptic points. The method is an updated form of a differential construction of the elliptic modular function $j$, first performed by Dedekind in 1877. Subtle differences between automorphic functions with respect to conjugate Fuchsian groups become apparent