Rosa Maria Miró-Roig - Böcker

This book presents progress on two open problems within the framework of algebraic geometry and commutative algebra: Gröbner's problem regarding the arithmetic Cohen-Macaulayness (aCM) of projections of Veronese varieties, and the problem of determining the structure of the algebra of invariants of finite groups.

The Third European Congress of Mathematics (3ecm) took place from July 10th to July 14th, 2000 in Barcelona. It was organised by the Societat Catalana de Matematiques (Catalan Mathematical Society), under the auspices of the Euro- pean Mathematical Society (EMS). As foreseen by the EMS and the International Mathematical Union, this Congress was a major event in World Mathematical Year 2000. In addition to reviewing outstanding research achievements, important aspects of the life of European mathematics were discussed. Mathematics is undergoing a period of rapid changes. Effective computation and applications in science and technology go ever more hand in hand with con- ceptual developments. It was one of the aims of 3ecm to reflect this mutual enrich- ment, while steering present and future trends of mathematical sciences. In fact, the motto of the Congress, Shaping the 21st Century, was meant to capture these views.Nearly 1400 people from 87 countries gathered together in the Palau de Con- gressos of Barcelona in order to take part in the activities of the 3ecm scientific programme: Nine plenary lectures, thirty invited lectures in parallel sessions, lec- tures given by EMS prize winners, ten mini-symposia on special topics, seven round tables, poster sessions, presentations of mathematical software and video exhibitions. Twenty events were satellites of 3ecm in various countries.

This volume contains refereed papers related to the lectures and talks given at a conference held in Siena (Italy) in June 2004. Also included are research papers that grew out of discussions among the participants and their collaborators. All the papers are research papers, but some of them also contain expository sections which aim to update the state of the art on the classical subject of special projective varieties and their applications and new trends like phylogenetic algebraic geometry.The topic of secant varieties and the classification of defective varieties is central and ubiquitous in this volume. Besides the intrinsic interest of the subject, it turns out that it is also relevant in other fields of mathematics like expressions of polynomials as sums of powers, polynomial interpolation, rank tensor computations, Bayesian networks, algebraic statistics and number theory.

The goal of this book is to cover the active developments of arithmetically Cohen-Macaulay and Ulrich bundles and related topics in the last 30 years, and to present relevant techniques and multiple applications of the theory of Ulrich bundles to a wide range of problems in algebraic geometry as well as in commutative algebra.