Algebraic Geometry is a deep subject of remarkable power that originated in the study of concrete objects, as it is concerned with the geometry of solutions to polynomial equations. It is this interplay between theoretical abstraction and tangible examples that gives the subject relevance and a reach into other areas of mathematics and its applications. There is no better subdiscipline of algeb...

We begin with a short (obviously incomplete) historical summary related to the book being reviewed. A complex number α is said to be “algebraic” if there is a polynomial 0 6= p(x) with integer coefficients such that p(α) = 0; otherwise α is said to be “transcendental”. The first recorded instance of mathematicians encountering an “irrational” algebraic number appears to be the ancient Greeks. I...

It took me more than a decade to come to terms with this result [that simple computer programs can produce complex behavior], and to realize just how fundamental and far-reaching its consequences are. In retrospect there is no reason the result could not have been found centuries ago, but increasingly I have come to view it as one of the more important single discoveries in the whole history of...

Representations of semisimple Lie algebras in the BGG category í µí²ª, by James E.

Broadly speaking, one of the primary tasks in the theory of compact Riemann surfaces is the construction of meromorphic functions. A powerful tool in this endeavor is the Riemann-Roch theorem which allows the construction of functions with partially prescribed zeros and poles, such that the number of prescribed poles exceeds the zeros by a fixed constant depending on the surface. To be more pre...

The discovery 18 years ago by Vaughan Jones of a powerful new polynomial invariant for knots and links began a revolution in knot theory. Of the nearly 8,000 Mathematical Reviews about knots, more than two thirds have appeared after 1985. Connections between knot theory and other areas of mathematics as well as physics are no longer surprising. Into this heady environment come two new publicati...

Dengue fever is a viral mosquito-borne infection, a major international public health concern with more than 55% of the world population at risk of acquiring the infection. Two variants of the disease exist: dengue fever (DF), a non-fatal form of illness, and dengue hemorrhagic fever (DHF), which may evolve toward a severe form known as dengue shock syndrome (DSS). Epidemiological studies suppo...

In this article, I discuss material which is related to the recent proof of Fermat’s Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey’s construction, and the conjectures of Serre and of Taniyama-Shimura.

Suppose that p, u, v, and w are integers, with p > 1. If u + v + w = 0, then uvw = 0. Professor Andrew Wiles of Princeton University deduced this form of Fermat’s Last Theorem at the conclusion of a series of three lectures during the June, 1993 workshop on Iwasawa theory, autmorphic forms, and p-adic representations at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK. Wile...