A. Polishchuk, University of Oregon, Eugene, OR,
and L. Positselski, Independent University of Moscow, Russia

Quadratic Algebras

Expected publication date is December 16, 2005

Description

This book introduces recent developments in the study of algebras defined by quadratic relations. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, $K$-theory, number theory, and noncommutative linear algebra.

The authors give a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincare-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.

The book can be used by graduate students and researchers working in algebra and any of the above-mentioned areas of mathematics.

Contents

Preliminaries
Koszul algebras and modules
Operations on graded algebras and modules
Poincare-Birkhoff-Witt bases
Nonhomogeneous quadratic algebras
Families of quadratic algebras and Hilbert series
Hilbert series of Koszul algebras and one-dependent processes
DG-algebras and Massey products
Bibliography

Details:

Series: University Lecture Series,Volume: 37
Publication Year: 2005
ISBN: 0-8218-3834-2
Paging: approximately 176 pp.
Binding: Softcover

Sean Dineen, University College Dublin, Ireland

Probability Theory in Finance:
A Mathematical Guide to the Black-Scholes Formula

Expected publication date is December 15, 2005

Description

The use of the Black-Scholes model and formula is pervasive in financial markets. There are very few undergraduate textbooks available on the subject and, until now, almost none written by mathematicians. Based on a course given by the author, the goal of this book is to introduce advanced undergraduates and beginning graduate students studying the mathematics of finance to the Black-Scholes formula. The author uses a first-principles approach, developing only the minimum background necessary to justify mathematical concepts and placing mathematical developments in context.

The book skillfully draws the reader toward the art of thinking mathematically and then proceeds to lay the foundations in analysis and probability theory underlying modern financial mathematics. It rigorously reveals the mathematical secrets of topics such as abstract measure theory, conditional expectations, martingales, Wiener processes, the Ito calculus, and other ingredients of the Black-Scholes formula. In explaining these topics, the author uses examples drawn from the universe of finance. The book also contains many exercises, some included to clarify simple points of exposition, others to introduce new ideas and techniques, and a few containing relatively deep mathematical results.

With the modest prerequisite of a first course in calculus, the book is suitable for undergraduates and graduate students in mathematics, finance, and economics and can be read, using appropriate selections, at a number of levels.

Contents

Money and markets
Fair games
Set theory
Measurable functions
Probability spaces
Expected values
Continuity and integrability
Conditional expectation
Martingales
The Black-Scholes formula
Stochastic integration
Solutions
Bibliography
Index

Details:

Series: Graduate Studies in Mathematics, Volume: 70
Publication Year: 2005
ISBN: 0-8218-3951-9
Paging: 294 pp.
Binding: Hardcover

Antonio Giambruno, Universita di Palermo, Italy,
and Mikhail Zaicev, Moscow State University, Russia

Polynomial Identities and Asymptotic Methods

Expected publication date is December 7, 2005

Description

This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory.

A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent. Results are extended to graded algebras and algebras with involution.

The book concludes with a study of the numerical invariants and their asymptotics in the class of Lie algebras. Even in algebras that are close to being associative, the behavior of the sequences of codimensions can be wild.

The material is suitable for graduate students and research mathematicians interested in polynomial identity algebras.

Contents

Polynomial identities and PI-algebras
$S_n$-representations
Group gradings and group actions
Codimension and colength growth
Matrix invariants and central polynomials
The PI-exponent of an algebra
Polynomial growth and low PI-exponent
Classifying minimal varieties
Computing the exponent of a polynomial
$G$-identities and $G\wr S_n$-action
Superalgebras, *-algebras and codimension growth
Lie algebras and non-associative algebras
The generalized-six-square theorem
Bibliography
Index

Details:

Series: Mathematical Surveys and Monographs, Volume: 122
Publication Year: 2005
ISBN: 0-8218-3829-6
Paging: 352 pp.
Binding: Hardcover

Barbara Fantechi, SISSA, Trieste, Italy, Lothar Gottsche, International Centre for Theoretical Physics, Trieste, Italy, Luc Illusie, Universite Paris-Sud, Orsay, France, Steven L. Kleiman, MIT, Cambridge, MA, Nitin Nitsure, Tata Institute of Fundamental Research, Mumbai, India, and Angelo Vistoli, Universita di Bologna, Italy

Fundamental Algebraic Geometry: Grothendieck's FGA Explained


Expected publication date is December 31, 2005

Description

Alexander Grothendieck's concepts turned out to be astoundingly powerful and productive, truly revolutionizing algebraic geometry. He sketched his new theories in talks given at the Seminaire Bourbaki between 1957 and 1962. He then collected these lectures in a series of articles in Fondements de la geometrie algebrique (commonly known as FGA).

Much of FGA is now common knowledge. However, some of it is less well known, and only a few geometers are familiar with its full scope. The goal of the current book, which resulted from the 2003 Advanced School in Basic Algebraic Geometry (Trieste, Italy), is to fill in the gaps in Grothendieck's very condensed outline of his theories. The four main themes discussed in the book are descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. The authors present complete proofs of the main results, using newer ideas to promote understanding whenever necessary, and drawing connections to later developments.

With the main prerequisite being a thorough acquaintance with basic scheme theory, this book is a valuable resource for anyone working in algebraic geometry.

Contents
Grothendieck topologies, fibered categories and descent theory
Introduction
Preliminary notions
Contravariant functors
Fibered categories
Stacks
Construction of Hilbert and Quot schemes
Construction of Hilbert and Quot schemes
Local properties and Hilbert scheme of points
Introduction
Elementary deformation theory
Hilbert schemes of points
Grothendieck's existence theorem in formal geometry
Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre
The Picard scheme
The Picard scheme
Bibliography
Index

Details:

Series: Mathematical Surveys and Monographs,Volume: 123
Publication Year: 2005
ISBN: 0-8218-3541-6
Paging: approximately 352 pp.
Binding: Softcover


Frank Morgan, Williams College, Williamstown, MA

Real Analysis and Applications:
Including Fourier Series and the Calculus of Variations

Expected publication date is January 1, 2006

Description

Real Analysis and Applications starts with a streamlined, but complete, approach to real analysis. It finishes with a wide variety of applications in Fourier series and the calculus of variations, including minimal surfaces, physics, economics, Riemannian geometry, and general relativity. The basic theory includes all the standard topics: limits of sequences, topology, compactness, the Cantor set and fractals, calculus with the Riemann integral, a chapter on the Lebesgue theory, sequences of functions, infinite series, and the exponential and Gamma functions. The applications conclude with a computation of the relativistic precession of Mercury's orbit, which Einstein called "convincing proof of the correctness of the theory [of General Relativity]."

The text not only provides clear, logical proofs, but also shows the student how to derive them. The excellent exercises come with select solutions in the back. This is a text that makes it possible to do the full theory and significant applications in one semester.

Frank Morgan is the author of six books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this applied version of his Real Analysis text, Morgan brings his famous direct style to the growing numbers of potential mathematics majors who want to see applications along with the theory.

The book is suitable for undergraduates interested in real analysis.

Contents

Part I: Real numbers and limits
Numbers and logic
Infinity
Sequences
Subsequences
Functions and limits
Composition of functions
Part II: Topology
Open and closed sets
Compactness
Existence of maximum
Uniform continuity
Connected sets and the intermediate value theorem
The Cantor set and fractals
Part III: Calculus
The derivative and the mean value theorem
The Riemann integral
The fundamental theorem of calculus
Sequences of functions
The Lebesgue theory
Infinite series $\sum_{n=1}^\infty a_n$
Absolute convergence
Power series
The exponential functions
Volumes of $n$-balls and the gamma function
Part IV: Fourier series
Fourier series
Strings and springs
Convergence of Fourier series
Part V: The calculus of variations
Euler's equation
First integrals and the Brachistochrone problem
Geodesics and great circles
Variational notation, higher order equations
Harmonic functions
Minimal surfaces
Hamilton's action and Lagrange's equations
Optimal economic strategies
Utility of consumption
Riemannian geometry
Noneuclidean geometry
General relativity
Partial solutions to exercises
Greek letters
Index

Details:

Publication Year: 2005
ISBN: 0-8218-3841-5
Paging: approximately 208 pp.
Binding: Hardcover