Often mathematicians refer to a "beautiful" result or a "beautiful" proof. In this special lecture, Enrico Bombieri, Professor Emeritus in the School of Mathematics, addresses the question, "What is beauty in mathematics?"

A `toy model' for studying the probabilistic distribution of nodal curves of eigenfunctions of linear operators arises from the Laplacian on the standard real 2-torus. Here the eigenvalues are associate to integers m that are sum of two squares, with multiplicity equal to the number of such representations. When the number of representations increases to infinity, it makes sense to consider the associated random eigenfunctions. The calculation of the variance is crucial and leads to the problem which is the object of this talk.

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In this lecture, we study the Moebius inversion for functions of this type and show how its behavior is related to the distribution of zeros in the half-plane of absolute convergence. Work in collaboration with Amit Ghosh.

In this lecture, Enrico Bombieri, IBM von Neumann Professor in the School of Mathematics, attempts to give an idea of the numerous different notions of truth in mathematics. Using accessible examples, he explains the difference between truth, proof, and verification. Bombieri, one of the world’s leading authorities on number theory and analysis, was awarded the Fields Medal in 1974 for his work on the large sieve and its application to the distribution of prime numbers. Some of his work has potential practical applications to cryptography and security of data transmission and identification.

This lecture by Enrico Bombieri, IBM von Neumann Professor in the School of Mathematics, explores how mathematics has arrived at its present pragmatic view of infinity and some of the counterintuitive paradoxes, as well as some of the positive results, deriving from its acceptance. It concludes with a view of how computer science is leading today to a new precise concept, namely the impossibly large in the realm of the finite.