Fields Institute Communications, Volume: 58
2010; 133 pp; hardcover
ISBN-13: 978-0-8218-4311-6
Expected publication date is December 12, 2010.
It is by now a well-known paradigm that public-key cryptosystems can be
built using finite Abelian groups and that algebraic geometry provides
a supply of such groups through Abelian varieties over finite fields. Of
special interest are the Abelian varieties that are Jacobians of algebraic
curves. All of the articles in this volume are centered on the theme of
point counting and explicit arithmetic on the Jacobians of curves over
finite fields. The topics covered include Schoof's ell-adic point counting
algorithm, the p-adic algorithms of Kedlaya and Denef-Vercauteren, explicit
arithmetic on the Jacobians of C_{ab} curves and zeta functions.
This volume is based on seminars on algebraic curves and cryptography held at the GANITA Lab of the University of Toronto during 2001-2008. The articles are mostly suitable for independent study by graduate students who wish to enter the field, both in terms of introducing basic material as well as guiding them in the literature. The literature in cryptography seems to be growing at an exponential rate. For a new entrant into the subject, navigating through this ocean can seem quite daunting. In this volume, the reader is steered toward a discussion of a few key ideas of the subject, together with some brief guidance for further reading. It is hoped that this approach may render the subject more approachable.
Graduate students and research mathematicians interested in cryptography, applications of number theory and algebraic geometry.
Mathematical Surveys and Monographs, Volume: 167
2010; 246 pp; hardcover
ISBN-13: 978-0-8218-4972-9
Expected publication date is December 3, 2010.
Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry.
This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.
Research mathematicians interested in cluster algebras and applications to geometry.
Mathematical Surveys and Monographs,Volume: 168
2010; 411 pp; hardcover
ISBN-13: 978-0-8218-5262-0
Expected publication date is November 28, 2010.
The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras.
The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel-Bernstein-Gelfand-Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form.
The next two chapters are also "primers" on coalgebras and Hopf algebras, respectively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4-7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics.
Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibliography.
Mathematical Surveys and Monographs, Volume: 169
2010; 318 pp; hardcover
ISBN-13: 978-0-8218-5189-0
Expected publication date is November 24,
This book is about equivariant real and complex topological K-theory for
finite groups. Its main focus is on the study of real connective K-theory
including ko^*(BG) as a ring and ko_*(BG) as a module over it. In the course
of their study the authors define equivariant versions of connective KO-theory
and connective K-theory with reality, in the sense of Atiyah, which give
well-behaved, Noetherian, uncompleted versions of the theory. They prove
local cohomology and completion theorems for these theories, giving a means
of calculation as well as establishing their formal credentials. In passing
from the complex to the real theories in the connective case, the authors
describe the known failure of descent and explain how the eta-Bockstein
spectral sequence provides an effective substitute.
This formal framework allows the authors to give a systematic calculation
scheme to quantify the expectation that ko^*(BG) should be a mixture of
representation theory and group cohomology. It is characteristic that this
starts with ku^*(BG) and then uses the local cohomology theorem and the
Bockstein spectral sequence to calculate ku_*(BG), ko^*(BG), and ko_*(BG).
To give the skeleton of the answer, the authors provide a theory of ko-characteristic
classes for representations, with the Pontrjagin classes of quaternionic
representations being the most important.
Building on the general results, and their previous calculations, the authors
spend the bulk of the book giving a large number of detailed calculations
for specific groups (cyclic, quaternion, dihedral, A_4, and elementary
abelian 2-groups). The calculations illustrate the richness of the theory
and suggest many further lines of investigation. They have been applied
in the verification of the Gromov-Lawson-Rosenberg conjecture for several
new classes of finite groups.
Graduate students and research mathematicians interested in connective K-theory.
Contemporary Mathematics, Volume: 527
2010; 168 pp; softcover
ISBN-13: 978-0-8218-4884-5
This volume contains contributions from the NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry, which was held from December 13-17, 2008 at Kansas State University in Manhattan, Kansas.
It gives an excellent picture of numerous connections of mirror symmetry with other areas of mathematics (especially with algebraic and symplectic geometry) as well as with other areas of mathematical physics. The techniques and methods used by the authors of the volume are at the frontier of this very active area of research.
Graduate students and research mathematicians interested in mirror symmetry.