Akira Kono and Dai Tamaki
Generalized Cohomology
Description
In the 1950s, Eilenberg and Steenrod presented their famous
characterization of homology theory by seven axioms. Somewhat
later, it was found that keeping just the first six of these
axioms (all except the condition on the "homology" of
the point), one can obtain many other interesting systems of
algebraic invariants of topological manifolds, such as $K$-theory,
cobordisms, and others. These theories come under the common name
of generalized homology (or cohomology) theories.
The purpose of the book is to give an exposition of generalized (co)homology
theories that can be read by a wide group of mathematicians who
are not experts in algebraic topology. It starts with basic
notions of homotopy theory and then introduces the axioms of
generalized (co)homology theory. Then the authors discuss various
types of generalized cohomology theories, such as complex-oriented
cohomology theories and Chern classes, $K$-theory, complex
cobordisms, and formal group laws. A separate chapter is devoted
to spectral sequences and their use in generalized cohomology
theories.
The book is intended to serve as an introduction to the subject
for mathematicians who do not have advanced knowledge of
algebraic topology. Prerequisites include standard graduate
courses in algebra and topology, with some knowledge of ordinary
homology theory and homotopy theory.
Readership
Graduate students and research mathematicians interested in
algebraic topology.
Contents
Preliminaries
Generalized cohomology
Characteristic classes of vector bundles
$K$-theory
Spectral sequence
Complex cobordism and its applications
Simplicial techniques
Limits
Spectrum
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs Volume: 230
Publication Year: 2006
ISBN: 0-8218-3514-9
Paging: approximately 272 pp.
Binding: Softcover
Dudley E. Littlewood
The Theory of Group Characters and Matrix Representations of
Groups: Second Edition
Description
Originally written in 1940, this book remains a classical source
on representations and characters of finite and compact groups.
The book starts with necessary information about matrices,
algebras, and groups. Then the author proceeds to representations
of finite groups. Of particular interest in this part of the book
are several chapters devoted to representations and characters of
symmetric groups and the closely related theory of symmetric
polynomials. The concluding chapters present the representation
theory of classical compact Lie groups, including a detailed
description of representations of the unitary and orthogonal
groups. The book, which can be read with minimal prerequisites (an
undergraduate algebra course), allows the reader to get a good
understanding of beautiful classical results about group
representations.
Readership
Graduate students and research mathematicians interested in
representation theory.
Contents
Matrices
Algebras
Groups
The Frobenius algebra
The symmetric group
Immanants and $S$-functions
$S$-functions of special series
The calculation of the characters of the symmetric group
Group characters and the structure of groups
Continuous matrix groups and invariant matrices
Groups of unitary matrices
Appendix
Bibliography
Supplementary bibliography
Index
Details:
Series: AMS Chelsea Publishing
Publication Year: 2006
ISBN: 0-8218-4067-3
Paging: 310 pp.
Binding: Hardcover
John McCleary
A First Course in Topology: Continuity and Dimension
Description
How many dimensions does our universe require for a comprehensive
physical description? In 1905, Poincare argued philosophically
about the necessity of the three familiar dimensions, while
recent research is based on 11 dimensions or even 23 dimensions.
The notion of dimension itself presented a basic problem to the
pioneers of topology. Cantor asked if dimension was a topological
feature of Euclidean space. To answer this question, some
important topological ideas were introduced by Brouwer, giving
shape to a subject whose development dominated the twentieth
century.
The basic notions in topology are varied and a comprehensive
grounding in point-set topology, the definition and use of the
fundamental group, and the beginnings of homology theory requires
considerable time. The goal of this book is a focused
introduction through these classical topics, aiming throughout at
the classical result of the Invariance of Dimension.
This text is based on the author's course given at Vassar College
and is intended for advanced undergraduate students. It is
suitable for a semester-long course on topology for students who
have studied real analysis and linear algebra. It is also a good
choice for a capstone course, senior seminar, or independent
study.
Readership
Undergraduate and Graduate Students interested in Topology.
Contents
A little set theory
Metric and topological spaces
Geometric notions
Building new spaces from old
Connectedness
Compactness
Homotopy and the fundamental group
Computations and covering spaces
The Jordan Curve Theorem
Simplicial complexes
Homology
Bibliography
Details:
Series: Student Mathematical Library,Volume: 31
Publication Year: 2006
ISBN: 0-8218-3884-9
Paging: approximately 216 pp.
Binding: Softcover
R. J. Williams, University of California, San Diego, La Jolla, California, USA
Introduction to the Mathematics of Finance
Description
The modern subject of mathematical finance has undergone
considerable development, both in theory and practice, since the
seminal work of Black and Scholes appeared a third of a century
ago. This book is intended as an introduction to some elements of
the theory that will enable students and researchers to go on to
read more advanced texts and research papers.
The book begins with the development of the basic ideas of
hedging and pricing of European and American derivatives in the
discrete (i.e., discrete time and discrete state) setting of
binomial tree models. Then a general discrete finite market model
is introduced, and the fundamental theorems of asset pricing are
proved in this setting. Tools from probability such as
conditional expectation, filtration, (super)martingale,
equivalent martingale measure, and martingale representation are
all used first in this simple discrete framework. This provides a
bridge to the continuous (time and state) setting, which requires
the additional concepts of Brownian motion and stochastic
calculus. The simplest model in the continuous setting is the
famous Black-Scholes model, for which pricing and hedging of
European and American derivatives are developed. The book
concludes with a description of the fundamental theorems for a
continuous market model that generalizes the simple Black-Scholes
model in several directions.
Readership
Graduate Students interested in financial mathematics.
Contents
Financial markets and derivatives
Binomial model
Finite market model
Black-Scholes model
Multi-dimensional Black-Scholes model
Conditional expectation and $L^p$-spaces
Discrete time stochastic processes
Continuous time stochastic processes
Brownian motion and stochastic integration
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics, Volume: 72
Publication Year: 2006
ISBN: 0-8218-3903-9
Paging: 152 pp.
Binding: Hardcover