Recent Advances in Algebraic Geometry

Author: Christopher D. Hacon,Mircea Mustaţă,Mihnea Popa

Publisher: Cambridge University Press

ISBN: 110764755X

Category: Mathematics

Page: 447

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A comprehensive collection of expository articles on cutting-edge topics at the forefront of research in algebraic geometry.

Foliation Theory in Algebraic Geometry

Author: Paolo Cascini,James McKernan,Jorge Vitório Pereira

Publisher: Springer

ISBN: 3319244604

Category: Mathematics

Page: 216

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Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classification of psuedoeffective codimension one distributions. Foliations play a fundamental role in algebraic geometry, for example in the proof of abundance for threefolds and to a solution of the Green-Griffiths conjecture for surfaces of general type with positive Segre class. The purpose of this volume is to foster communication and enable interactions between experts who work on holomorphic foliations and birational geometry, and to bring together leading researchers to demonstrate the powerful connection of ideas, methods, and goals shared by these two areas of study./div

Recent Advances in Hodge Theory

Period Domains, Algebraic Cycles, and Arithmetic

Author: Matt Kerr,Gregory Pearlstein

Publisher: Cambridge University Press

ISBN: 110754629X

Category: Mathematics

Page: 528

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Combines cutting-edge research and expository articles in Hodge theory. An essential reference for graduate students and researchers.

Synthetic Differential Geometry

Author: Anders Kock

Publisher: Cambridge University Press

ISBN: 0521687381

Category: Mathematics

Page: 233

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Second edition of this book detailing how limit processes can be represented algebraically.

Advances in Elliptic Curve Cryptography

Author: Ian F. Blake,Gadiel Seroussi,Nigel P. Smart

Publisher: Cambridge University Press

ISBN: 9781139441223

Category: Mathematics

Page: N.A

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Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular. Other topics such as the Weil and Tate pairings have been applied in new and important ways to cryptographic protocols that hold great promise. Notions such as provable security, side channel analysis and the Weil descent technique have also grown in importance. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important topics. They are divided into the areas of protocols, implementation techniques, mathematical foundations and pairing based cryptography. Each of the topics is presented in an accessible, coherent and consistent manner for a wide audience that will include mathematicians, computer scientists and engineers.

Two-Dimensional Homotopy and Combinatorial Group Theory

Author: Cynthia Hog-Angeloni,Wolfgang Metzler,Allan J. Sieradski

Publisher: Cambridge University Press

ISBN: 9780521447003

Category: Mathematics

Page: 412

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Basic work on two-dimensional homotopy theory dates back to K. Reidemeister and J. H. C. Whitehead. Much work in this area has been done since then, and this book considers the current state of knowledge in all the aspects of the subject. The editors start with introductory chapters on low-dimensional topology, covering both the geometric and algebraic sides of the subject, the latter including crossed modules, Reidemeister-Peiffer identities, and a concrete and modern discussion of Whitehead's algebraic classification of 2-dimensional homotopy types. Further chapters have been skilfully selected and woven together to form a coherent picture. The latest algebraic results and their applications to 3- and 4-dimensional manifolds are dealt with. The geometric nature of the subject is illustrated to the full by over 100 diagrams. Final chapters summarize and contribute to the present status of the conjectures of Zeeman, Whitehead, and Andrews-Curtis. No other book covers all these topics. Some of the material here has been used in courses, making this book valuable for anyone with an interest in two-dimensional homotopy theory, from graduate students to research workers.

Topological Methods in Group Theory

Author: Ross Geoghegan

Publisher: Springer Science & Business Media

ISBN: 0387746110

Category: Mathematics

Page: 473

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This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.

The Grothendieck Theory of Dessins D'Enfants

Author: Leila Schneps

Publisher: Cambridge University Press

ISBN: 9780521478212

Category: Mathematics

Page: 368

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Dessins d'Enfants are combinatorial objects, namely drawings with vertices and edges on topological surfaces. Their interest lies in their relation with the set of algebraic curves defined over the closure of the rationals, and the corresponding action of the absolute Galois group on them. The study of this group via such realted combinatorial methods as its action on the Dessins and on certain fundamental groups of moduli spaces was initiated by Alexander Grothendieck in his unpublished Esquisse d'un Programme, and developed by many of the mathematicians who have contributed to this volume. The various articles here unite all of the basics of the subject as well as the most recent advances. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.

Arithmetic of Blowup Algebras

Author: Wolmer V. Vasconcelos

Publisher: Cambridge University Press

ISBN: 9780521454841

Category: Mathematics

Page: 329

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This book provides an introduction to recent developments in the theory of blow up algebras - Rees algebras, associated graded rings, Hilbert functions, and birational morphisms. The emphasis is on deriving properties of rings from their specifications in terms of generators and relations. While this limits the generality of many results, it opens the way for the application of computational methods. A highlight of the book is the chapter on advanced computational methods in algebra using Gröbner basis theory and advanced commutative algebra. The author presents the Gröbner basis algorithm and shows how it can be used to resolve computational questions in algebra. This volume is intended for advanced students in commutative algebra, algebraic geometry and computational algebra, and homological algebra. It can be used as a reference for the theory of Rees algebras and related topics.

O-Minimality and Diophantine Geometry

Author: A. J. Wilkie,G. O. Jones

Publisher: Cambridge University Press

ISBN: 1107462495

Category: Mathematics

Page: 232

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Brings the researcher up to date with recent applications of mathematical logic to number theory.

Geometric and Cohomological Group Theory

Author: Peter H. Kropholler,Ian J. Leary,Conchita Martínez-Pérez,Brita E. A. Nucinkis

Publisher: Cambridge University Press

ISBN: 131662322X

Category: Mathematics

Page: 278

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Surveys the state of the art in geometric and cohomological group theory. Ideal entry point for young researchers.

Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics

Author: Pramod M. Achar,Dijana Jakelić,Kailash C. Misra,Milen Yakimov

Publisher: American Mathematical Society

ISBN: 0821898523

Category: Mathematics

Page: 280

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This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac—Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.

Geometry of Low-dimensional Manifolds

Proceedings of the Durham Symposium, July 1989

Author: S. K. Donaldson,Charles Benedict Thomas

Publisher: Cambridge University Press

ISBN: 9780521400015

Category: Low-dimensional topology

Page: 242

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Surveys on Recent Developments in Algebraic Geometry

Author: Izzet Coskun,Tommaso de Fernex,Angela Gibney

Publisher: American Mathematical Soc.

ISBN: 1470435578

Category: $K$-theory -- Higher algebraic $K$-theory -- $Q$- and plus-constructions

Page: 370

View: 3281

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The algebraic geometry community has a tradition of running a summer research institute every ten years. During these influential meetings a large number of mathematicians from around the world convene to overview the developments of the past decade and to outline the most fundamental and far-reaching problems for the next. The meeting is preceded by a Bootcamp aimed at graduate students and young researchers. This volume collects ten surveys that grew out of the Bootcamp, held July 6–10, 2015, at University of Utah, Salt Lake City, Utah. These papers give succinct and thorough introductions to some of the most important and exciting developments in algebraic geometry in the last decade. Included are descriptions of the striking advances in the Minimal Model Program, moduli spaces, derived categories, Bridgeland stability, motivic homotopy theory, methods in characteristic and Hodge theory. Surveys contain many examples, exercises and open problems, which will make this volume an invaluable and enduring resource for researchers looking for new directions.

New Directions in Locally Compact Groups

Author: Pierre-Emmanuel Caprace,Nicolas Monod

Publisher: Cambridge University Press

ISBN: 1108351948

Category: Mathematics

Page: 358

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This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger–Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.

The Topology of Stiefel Manifolds

Author: I. M. James

Publisher: Cambridge University Press

ISBN: 0521213347

Category: Mathematics

Page: 168

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Stiefel manifolds are an interesting family of spaces much studied by algebraic topologists. These notes, which originated in a course given at Harvard University, describe the state of knowledge of the subject, as well as the outstanding problems. The emphasis throughout is on applications (within the subject) rather than on theory. However, such theory as is required is summarized and references to the literature are given, thus making the book accessible to non-specialists and particularly graduate students. Many examples are given and further problems suggested.

Complex Algebraic Curves

Author: Frances Clare Kirwan

Publisher: Cambridge University Press

ISBN: 9780521423533

Category: Mathematics

Page: 264

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This development of the theory of complex algebraic curves was one of the peaks of nineteenth century mathematics. They have many fascinating properties and arise in various areas of mathematics, from number theory to theoretical physics, and are the subject of much research. By using only the basic techniques acquired in most undergraduate courses in mathematics, Dr. Kirwan introduces the theory, observes the algebraic and topological properties of complex algebraic curves, and shows how they are related to complex analysis.