A Mathematical Gift, I The Interplay Between Topology, Functions, Geometry, and Algebra

This is the first of three volumes originated from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students. The main purpose of the lectures was to show the listeners the beauty and liveliness of mathematics using the material that is accessible to people with little preliminary knowledge. The first chapter of the book talks about the geometry and topology of surfaces. Among other topics the authors discuss the Poincare-Hopf theorem about critical points of vector fields on surfaces and the Gauss-Bonnet theorem about the relation between the curvature and topology (Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach to dimension. It also discusses the structure of three-dimensional manifolds, proving, in particular, that the three-dimensional sphere is the union of two doughnuts.