This volume brings together recent, original research and survey articles by leading experts in several fields that include singularity theory, algebraic geometry and commutative algebra. The motivation for this collection comes from the wide-ranging research of the distinguished mathematician, Antonio Campillo, in these and related fields. Besides his influence in the mathematical community stemming from his research, Campillo has also endeavored to promote mathematics and mathematicians' networking everywhere, especially in Spain, Latin America and Europe. Because of his impressive achievements throughout his career, we dedicate this book to Campillo in honor of his 65th birthday. Researchers and students from the world-wide, and in particular Latin American and European, communities in singularities, algebraic geometry, commutative algebra, coding theory, and other fields covered in the volume, will have interest in this book.

This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves. While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category. The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.

This volume consists of ten articles which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject. This is the first volume in a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

This volume contains the proceedings of the ICM 2018 satellite school and workshop K-theory conference in Argentina. The school was held from July 16–20, 2018, in La Plata, Argentina, and the workshop was held from July 23–27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in K-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as K-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative K-groups of monoid algebras, Milnor K-theory and regulators, noncommutative motives, the classification of C∗-algebras via Kasparov's K-theory, the comparison between full and reduced C∗-crossed products, and a proof of Bott periodicity using almost commuting matrices.

This book gives an overview of research in the topology and geometry of intersections of quadrics in $\mathbb{R}^n$, with a focus on intersections of concentric ellipsoids and related spaces. Unifying and organizing material previously spread over many articles, it also contains new results. The first part provides very detailed foundations of a wide-ranging theory that could be useful for future developments. It includes chapters on general intersections of quadrics, operations on them, and intersections of concentric and coaxial quadrics. Moving from the general to the specific, the second part focuses on a topological description of transverse intersections of concentric ellipsoids, including a complete description of the case of three ellipsoids, and of some large families of more than three of them. The third part looks at relations to other areas of mathematics such as dynamical systems, complex geometry, contact and symplectic geometry, and other applications. An appendix gathers some technical items and also gives an account of the origins, motivations and progression of the subject, including historical recollections of the author, who has been central to its development.

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

This is the second volume of the Handbook of the Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. This volume consists of ten chapters which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory and related topics. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

This book, which explores recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, combinatorics, quantum algebras, and theoretical field, is conceived as a handbook to provide easy access to the present state of knowledge and stimulate further development. The many topics discussed include quivers, quivers with potential, bound quiver algebras, Jacobian algebras, cluster algebras and categories, Calabi-Yau algebras and categories, triangulated and derived categories, and quantum loop algebras. This book consists of thirteen self-contained expository survey and research articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. The articles contain a large number of examples and open problems and give new perspectives for research in the field.

This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, quantum groups, homological algebra, invariant theory, combinatorics, model theory and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include diagram algebras, Brauer algebras, cellular algebras, quasi-hereditary algebras, Hall algebras, Hecke algebras, symplectic reflection algebras, Cherednik algebras, Kashiwara crystals, Fock spaces, preprojective algebras, cluster algebras, rank varieties, varieties of algebras and modules, moduli of representations of quivers, semi-invariants of quivers, Cohen-Macaulay modules, singularities, coherent sheaves, derived categories, spectral representation theory, Coxeter polynomials, Auslander-Reiten theory, Calabi-Yau triangulated categories, Poincare duality spaces, selfinjective algebras, periodic algebras, stable module categories, Hochschild cohomologies, deformations of algebras, Galois coverings of algebras, tilting theory, algebras of small homological dimensions, representation types of algebras, and model theory. This book consists of fifteen self-contained expository survey articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of open problems and give new perspectives for research in the field.

This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 2 focuses on the most recent research.