A Course in Homological Algebra

Author: Peter J. Hilton,Urs Stammbach

Publisher: Springer Science & Business Media

ISBN: 1441985662

Category: Mathematics

Page: 366

View: 2124

Homological algebra has found a large number of applications in many fields ranging from finite and infinite group theory to representation theory, number theory, algebraic topology and sheaf theory. In the new edition of this broad introduction to the field, the authors address a number of select topics and describe their applications, illustrating the range and depth of their developments. A comprehensive set of exercises is included.

Lectures on Functor Homology

Author: Vincent Franjou,Antoine Touzé

Publisher: Birkhäuser

ISBN: 3319213059

Category: Mathematics

Page: 149

View: 6639

This book features a series of lectures that explores three different fields in which functor homology (short for homological algebra in functor categories) has recently played a significant role. For each of these applications, the functor viewpoint provides both essential insights and new methods for tackling difficult mathematical problems. In the lectures by Aurélien Djament, polynomial functors appear as coefficients in the homology of infinite families of classical groups, e.g. general linear groups or symplectic groups, and their stabilization. Djament’s theorem states that this stable homology can be computed using only the homology with trivial coefficients and the manageable functor homology. The series includes an intriguing development of Scorichenko’s unpublished results. The lectures by Wilberd van der Kallen lead to the solution of the general cohomological finite generation problem, extending Hilbert’s fourteenth problem and its solution to the context of cohomology. The focus here is on the cohomology of algebraic groups, or rational cohomology, and the coefficients are Friedlander and Suslin’s strict polynomial functors, a conceptual form of modules over the Schur algebra. Roman Mikhailov’s lectures highlight topological invariants: homoto py and homology of topological spaces, through derived functors of polynomial functors. In this regard the functor framework makes better use of naturality, allowing it to reach calculations that remain beyond the grasp of classical algebraic topology. Lastly, Antoine Touzé’s introductory course on homological algebra makes the book accessible to graduate students new to the field. The links between functor homology and the three fields mentioned above offer compelling arguments for pushing the development of the functor viewpoint. The lectures in this book will provide readers with a feel for functors, and a valuable new perspective to apply to their favourite problems.

Algebraic Geometry

Salt Lake City 2015 : 2015 Summer Research Institute, July 13-31, 2015, University of Utah, Salt Lake City, Utah

Author: Richard Thomas

Publisher: American Mathematical Soc.

ISBN: 1470435780

Category: Geometry, Algebraic

Page: 635

View: 5180

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Minimal Resolutions Via Algebraic Discrete Morse Theory

Author: Michael Jöllenbeck,Volkmar Welker

Publisher: American Mathematical Soc.

ISBN: 0821842579

Category: Mathematics

Page: 74

View: 347

Forman's discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoldberg, the authors show that this theory can be aplied to chain complexes of free modules over a ring and provide four applications of this theory.

An Algebraic Introduction to Complex Projective Geometry

Commutative Algebra

Author: Christian Peskine,Peskine Christian

Publisher: Cambridge University Press

ISBN: 9780521480727

Category: Mathematics

Page: 244

View: 2077

In this introduction to commutative algebra, the author choses a route that leads the reader through the essential ideas, without getting embroiled in technicalities. He takes the reader quickly to the fundamentals of complex projective geometry, requiring only a basic knowledge of linear and multilinear algebra and some elementary group theory. The author divides the book into three parts. In the first, he develops the general theory of noetherian rings and modules. He includes a certain amount of homological algebra, and he emphasizes rings and modules of fractions as preparation for working with sheaves. In the second part, he discusses polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, the author introduces affine complex schemes and their morphisms; he then proves Zariski's main theorem and Chevalley's semi-continuity theorem. Finally, the author's detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

Computer Algebra in Scientific Computing

CASC '99 : Proceedings of the Second Workshop on Computer Algebra in Scientific Computing, Munich, May 31-June 4, 1999

Author: Victor Grigorʹevich Ganzha,Ernst Mayr,Evgenii Vasilʹevich Vorozhtsov

Publisher: Springer Verlag

ISBN: 9783540660477

Category: Mathematics

Page: 509

View: 4249

This book contains papers submitted by the participants of the workshop on Computer Algebra in Scientific Computing CASC '99, as well as two invited papers. The collection of papers included in the proceedings covers various topics of computer algebra methods, algorithms and software applied to scientific computing. Moreover, applications of computer algebra methods for the solution of current problems in group theory are treated, which mostly arise in mathematical physics. Another important trend which may be seen from the present collection of papers is the application of computer algebra methods to the development of new efficient analytic and numerical solvers, both for ordinary and partial differential equations. Some papers deal with algorithmic and software aspects associated with the implementation of computer algebra methods, or study the stability of satellite and mechanical systems, or the application of computer algebra to the solution of problems in technology.