Monday, January 23 and Monday, January 30

CL 0171, 4:30-5:30

In this series of two talks we will examine how two problems which are very simple to state lead to some very difficult mathematics at the forefront of modern number theory. The first lecture will focus on the congruent number problem: "For which positive integers n is there a right triangle with rational sides and area n?" We will see how an investigation of this problem quickly leads into the land of elliptic curves and concludes with a conjecture worth a million dollars. The second problem we will explore is Fermat's last theorem. This states that for any integer n > 2 the only integers x, y, and z satisfying x^{n} + y^{n} = z^{n} must also satisfy xyz=0. Attempts to prove this theorem have led to much of modern number theory. We will use this as a basis for further study of elliptic curves, as well as a basis for introducing modular forms and Galois representations and the relationship between the three. It is hoped that you will come away from these talks with an appreciation for how aspects of algebra, analysis, and geometry all come together in modern number theory.