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Random matrix theory and number theory
Supervisor: Nina Snaith
Theme: Random Matrix
My main research interest is in random matrix theory and its application to number theory. Random matrix theory was developed in the context of nuclear physics to predict the statistics of the eigenenergies of complicated nuclear systems. However, in the 1970s, it was shown that the Riemann zeta function (a function much studied by number theorists and subject of the famous Riemann Hypothesis) has zeros that show the same statistics as these nuclear energy levels. So, random matrix theory has become a powerful tool to study number theoretical functions like the Riemann zeta function and other L-functions. Current projects in this area involve applying random matrix theory to mean values of more exotic L-functions, to the zero statistics of families of L-functions of interest in number theory, to the order of zeros of L-functions at the critical point (in connection with the conjecture of Birch and Swinnerton-Dyer) and in the distribution of prime numbers. Projects in this area can involve both numerical and analytical work and students will learn techniques that are of use both in mathematics and physics.