This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

A decade after the publication of Contemporary Mathematics Vol. 287, the present volume demonstrates the consolidation of important areas, such as algebraic statistics, computational commutative algebra, and deeper aspects of graphical models. --

"The second volume of the authors’ ‘Computational commutative algebra’...covers on its 586 pages a wealth of interesting material with several unexpected applications. ... an encyclopedia on computational commutative algebra, a source for lectures on the subject as well as an inspiration for seminars. The text is recommended for all those who want to learn and enjoy an algebraic tool that becomes more and more relevant to different fields of applications." --ZENTRALBLATT MATH

This is an introduction to some geometrie aspects of G-function theory. Most of the results presented here appear in print for the flrst time; hence this text is something intermediate between a standard monograph and a research artic1e; it is not a complete survey of the topic. Except for geometrie chapters (I.3.3, II, IX, X), I have tried to keep it reasonably self­ contained; for instance, the second part may be used as an introduction to p-adic analysis, starting from a few basic facts wh ich are recalled in IV.l.l. I have inc1uded about forty exercises, most of them giving some complements to the main text. Acknowledgements This book was written during a stay at the Max-Planck-Institut in Bonn. I should like here to express my special gratitude to this institute and its director, F. Hirzebruch, for their generous hospitality. G. Wüstholz has suggested the whole project and made its realization possible, and this book would not exist without his help; I thank him heartily. I also thank D. Bertrand, E. Bombieri, K. Diederich, and S. Lang for their encouragements, and D. Bertrand, G. Christo I and H Esnault for stimulating conversations and their help in removing some inaccuracies after a careful reading of parts of the text (any remaining error is however my sole responsibility).

The nature of interactions between mathematicians and physicists has been thoroughly transformed in recent years. String theory and quantum field theory have contributed a series of profound ideas that gave rise to entirely new mathematical fields and revitalized older ones. The influence flows in both directions, with mathematical techniques and ideas contributing crucially to major advances in string theory. A large and rapidly growing number of both mathematicians and physicists are working at the string-theoretic interface between the two academic fields. The String-Math conference series aims to bring together leading mathematicians and mathematically minded physicists working in this interface. This volume contains the proceedings of the inaugural conference in this series, String-Math 2011, which was held June 6-11, 2011, at the University of Pennsylvania.