2011; approx. 256 pp; softcover
ISBN-13: 978-0-8218-6914-7
Expected publication date is October 21, 2011.
One of the little-known effects of World War I was the collapse of international scientific cooperation. In mathematics, the discord continued after the war's end and after the Treaty of Versailles had been signed in 1919. Many distinguished scientists were involved in the war and its aftermath, and from their letters and papers, now almost a hundred years old, we learn of their anguished wartime views and their struggles afterwards either to prolong the schism in mathematics or to end it.
J. C. Fields, the foremost Canadian mathematician of his time, was educated in Canada, the United States, and Germany, and championed an international spirit of cooperation to further the frontiers of mathematics. It was during the awkward post-war period that J. C. Fields established the Fields Medal, an international prize for outstanding research, which soon became the highest award in mathematics. J. C. Fields intended it to be an international medal, and a glance at the varying backgrounds of the fifty-two Fields medallists shows it to be so.
Who was Fields? What carried him from Hamilton, Canada West, where he was born in 1863, into the middle of this turbulent era of international scientific politics? A modest mathematician, he was an unassuming man. This biography outlines Fields' life and times and the difficult circumstances in which he created the Fields Medal. It is the first such published study.
Anyone interested in the history of mathematics and specifically Fields, and the Fields medal.
Childhood of John Charles Fields
Toronto and Baltimore
Post-doctoral years in Europe, 1892-1900
Return to Canada
Fields and research
Mathematics before 1914: The golden years
Science responds to war
The politics of avoidance
International Mathematical Congress, Toronto 1924
"Sub-turbulent politics": Pincherle and Bologna
The Fields Medal
Late years
Publications of J. C. Fields
Fields Medallists, 1936-2010
Fields' colleagues and friends
Bibliography
Acknowledgments
Index
Mathematical Surveys and Monographs, Volume: 177
2011; approx. 413 pp; hardcover
ISBN-13: 978-0-8218-5360-3
Expected publication date is November 4, 2011.
This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory.
In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public key cryptography so far.
This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.
Graduate students and research mathematicians interested in the relations between group theory, cryptography, and complexity theory.
Introduction
Background on groups, complexity, and cryptography
Background on public key cryptography
Background on combinatorial group theory
Background on computational complexity
Non-commutative cryptography
Canonical non-commutative cryptography
Platform groups
More protocols
Using decision problems in public key cryptography
Authentication
Generic complexity and cryptanalysis
Distributional problems and the average case complexity
Generic case complexity
Generic complexity of NP-complete problems
Generic complexity of undecidable problems
Strongly, super, and absolutely undecidable problems
Asymptotically dominant properties and cryptanalysis
Asymptotically dominant properties
Length based and quotient attacks
Word and conjugacy search problems in groups
Word search problem
Conjugacy search problem
Word problem in some special classes of groups
Free solvable groups
Compressed words
Probabilistic group-based cryptanalysis
Bibliography
Abbreviations and notation
Index
Graduate Studies in Mathematics, Volume: 128
2011; approx. 438 pp; hardcover
ISBN-13: 978-0-8218-6907-9
Expected publication date is December 22, 2011.
Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language.
This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, mathbf{P} versus mathbf{NP}, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.
Graduate students and research mathematicians interested in tensors; researchers in the sciences and geometry.
Motivation from applications, multilinear algebra and elementary results
Introduction
Multilinear algebra
Elementary results on rank and border rank
Geometry and representation theory
Algebraic geometry for spaces of tensors
Secant varieties
Exploiting symmetry: Representation theory for spaces of tensors
Tests for border rank: Equations for secant varieties
Additional varieties useful for spaces of tensors
Rank
Normal forms for small tensors
Applications
The complexity of matrix multiplication
Tensor decomposition
mathbf{P} v. mathbf{NP}
Varieties of tensors in phylogenetics and quantum mechanics
Advanced topics
Overview of the proof of the Alexander-Hirschowitz theorem
Representation theory
Weyman's method
Hints and answers to selected exercises
Bibliography
Index
Graduate Studies in Mathematics, Volume: 127
2011; approx. 834 pp; hardcover
ISBN-13: 978-0-8218-5286-6
Expected publication date is January 15, 2012.
The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory.
This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.
Graduate students and research mathematicians interested in algebraic topology and homotopy theory.
The language of categories
Categories and functors
Limits and colimits
Semi-formal homotopy theory
Categories of spaces
Homotopy
Cofibrations and fibrations
Homotopy limits and colimits
Homotopy pushout and pullback squares
Tools and techniques
Topics and examples
Model categories
Four topological inputs
The concept of dimension in homotopy theory
Subdivision of disks
The local nature of fibrations
Pullbacks of cofibrations
Related topics
Targets as domains, domains as targets
Constructions of spaces and maps
Understanding suspension
Comparing pushouts and pullbacks
Some computations in homotopy theory
Further topics
Cohomology and homology
Cohomology
Homology
Cohomology operations
Chain complexes
Topics, problems and projects
Cohomology, homology and fibrations
The Wang sequence
Cohomology of filtered spaces
The Serre filtration of a fibration
Application: Incompressibility
The spectral sequence of a filtered space
The Leray-Serre spectral sequence
Application: Bott periodicity
Using the Leray-Serre spectral sequence
Vistas
Localization and completion
Exponents for homotopy groups
Classes of spaces
Miller's theorem
Some algebra
References
Index