He North Pole Project

The North Pole Project

Beginning Topology

Beginning Topology is designed to give undergraduate students a broad notion of the scope of topology in areas of point-set, geometric, combinatorial, differential, and algebraic topology, including an introduction to knot theory. A primary goal is to expose students to some recent research and to get them actively involved in learning. Exercises and open-ended projects are placed throughout the text, making it adaptable to seminar-style classes. The book starts with a chapter introducing the basic concepts of point-set topology, with examples chosen to captivate students' imaginations while illustrating the need for rigor. Most of the material in this and the next two chapters is essential for the remainder of the book. One can then choose from chapters on map coloring, vector fields on surfaces, the fundamental group, and knot theory. A solid foundation in calculus is necessary, with some differential equations and basic group theory helpful in a couple of chapters. Topics are chosen to appeal to a wide variety of students: primarily upper-level math majors, but also a few freshmen and sophomores as well as graduate students from physics, economics, and computer science. All students will benefit from seeing the interaction of topology with other fields of mathematics and science; some will be motivated to continue with a more in-depth, rigorous study of topology.

Topology of Numbers

This book serves as an introduction to number theory at the undergraduate level, emphasizing geometric aspects of the subject. The geometric approach is exploited to explore in some depth the classical topic of quadratic forms with integer coefficients, a central topic of the book. Quadratic forms of this type in two variables have a very rich theory, developed mostly by Euler, Lagrange, Legendre, and Gauss during the period 1750–1800. In this book their approach is modernized by using the splendid visualization tool introduced by John Conway in the 1990s called the topograph of a quadratic form. Besides the intrinsic interest of quadratic forms, this theory has also served as a stepping stone for many later developments in algebra and number theory. The book is accessible to students with a basic knowledge of linear algebra and arithmetic modulo $n$. Some exposure to mathematical proofs will also be helpful. The early chapters focus on examples rather than general theorems, but theorems and their proofs play a larger role as the book progresses.