Barry Mazur invites lovers of poetry to make a leap into mathematics. Through discussions of the role of the imagination and imagery in both poetry and mathematics, Mazur reviews the writings of the early mathematical explorers and reveals the early bafflement of these Renaissance thinkers faced with imaginary numbers. Then he shows us, step-by-step, how to begin imagining these strange mathematical objects ourselves.

The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology.

Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology.

...

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.

This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.