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

Non-linear finite WW-symmetries and applications in elementary systems

In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the theory finite $W$-algebras, which is an important class of non-linear symmetries. In particular, we discuss both the classical and quantum theory and elaborate on several aspects of their representation theory. Some new results are presented. These include finite $W$ coadjoint orbits, real forms and unitary representation of finite $W$-algebras and Poincare-Birkhoff-Witt theorems for finite $W$-algebras. Also we present some new finite $W$-algebras that are not related to $sl(2)$ embeddings. At the end of the paper we investigate how one could construct physical theories, for example gauge field theories, that are based on non-linear algebras.Comment: 88 pages, LaTe