Simple enough for detailed study, rich enough to show interesting behavior, K3 surfaces illuminate core methods in algebraic geometry.

This book provides an overview of the latest developments concerning the moduli of K3 surfaces. It is aimed at algebraic geometers, but is also of interest to number theorists and theoretical physicists, and continues the tradition of related volumes like “The Moduli Space of Curves” and “Moduli of Abelian Varieties,” which originated from conferences on the islands Texel and Schiermonnikoog and which have become classics. K3 surfaces and their moduli form a central topic in algebraic geometry and arithmetic geometry, and have recently attracted a lot of attention from both mathematicians and theoretical physicists. Advances in this field often result from mixing sophisticated techniques from algebraic geometry, lattice theory, number theory, and dynamical systems. The topic has received significant impetus due to recent breakthroughs on the Tate conjecture, the study of stability conditions and derived categories, and links with mirror symmetry and string theory. At the same time, the theory of irreducible holomorphic symplectic varieties, the higher dimensional analogues of K3 surfaces, has become a mainstream topic in algebraic geometry. Contributors: S. Boissière, A. Cattaneo, I. Dolgachev, V. Gritsenko, B. Hassett, G. Heckman, K. Hulek, S. Katz, A. Klemm, S. Kondo, C. Liedtke, D. Matsushita, M. Nieper-Wisskirchen, G. Oberdieck, K. Oguiso, R. Pandharipande, S. Rieken, A. Sarti, I. Shimada, R. P. Thomas, Y. Tschinkel, A. Verra, C. Voisin.

In recent years, research in K3 surfaces and Calabi–Yau varieties has seen spectacular progress from both arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics—in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds, held at the Fields Institute (August 16-25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi–Yau varieties. With the big variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated. Unlike most other conferences, the 2011 Calabi–Yau workshop started with 3 days of introductory lectures. A selection of 4 of these lectures is included in this volume. These lectures can be used as a starting point for the graduate students and other junior researchers, or as a guide to the subject.

Articles in this volume are based on lectures given at three conferences on Geometry at the Frontier, held at the Universidad de la Frontera, Pucón, Chile in 2016, 2017, and 2018. The papers cover recent developments on the theory of algebraic varieties—in particular, of their automorphism groups and moduli spaces. They will be of interest to anyone working in the area, as well as young mathematicians and students interested in complex and algebraic geometry.

Providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. It offers a comprehensive introduction to this fascinating topic, and will certainly become an essential reference for anybody working in the field. Rationality problems are of fundamental importance both in algebra and algebraic geometry. Historically, rationality problems motivated significant developments in the theory of abelian integrals, Riemann surfaces and the Abel–Jacobi map, among other areas, and they have strong links with modern notions such as moduli spaces, Hodge theory, algebraic cycles and derived categories. This text is aimed at researchers and graduate students in algebraic geometry.

Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results. The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side. Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.

This book, one of the first on G2 manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G2 manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G2 manifolds that have been recently discovered. The first half of the book, consisting of several introductory lectures, is aimed at experienced graduate students or early career researchers in geometry and topology who wish to familiarize themselves with this burgeoning field. The second half, consisting of numerous survey articles, is intended to be useful to both beginners and experts in the field.

It has been realized that Hilbert schemes originally studied in algebraic geometry are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory - even theoretical physics. This book reflects this feature of Hilbert schemes.

"The past year has witnessed truly remarkable developments in our understanding of string theory. Fields, Strings and Duality - TASI 96 is an invaluable collection of review papers on the subject, contributed by the most prominent researchers in the field. This volume is a scientific treasure for graduate students, researchers and all others who are interested in the progress of theoretical physics."--Publisher's website