About the Project
This project will be supervised by Dr Steve Lester.
L-functions and automorphic forms appear prominently in modern number theory and also arise in other areas of mathematics. The analytic properties of these functions are closely related to arithmetic problems involving the distribution of the prime numbers, lattice points, and structure of elliptic curves. For instance, Millennium prize problems such as the Generalised Riemann Hypothesis and Birch and Swinnerton-Dyer Conjecture are directly related to the theory of these functions.
This project aims to further understand the analytic behaviour of L-functions and automorphic forms, with applications to problems arising in number theory or mathematical physics. Specific problems will follow the interests of the student and possible topics include: the distribution of zeros of Hecke cusp forms, moment estimates for families of L-functions, and the behaviour of Fourier coefficients of automorphic forms.
Part of the project could potentially explore profound connections between $L$-functions and modular forms to questions in quantum chaos, which is an sub-field of mathematical physics which studies the quantum dynamics of classically chaotic systems. This is an especially exciting area of study that is still quite mysterious. Fascinatingly, much progress has been made in special arithmetic settings in which automrophic forms and $L$-functions play a prominent role.
Previous exposure to analytic number theory or,algebraic number theory would be advantageous, however not necessary and may be learned during PhD research. Background in real and complex analysis along with probability theory and harmonic analysis would be desirable.
The application procedure is described on the School website. For further inquiries please contact Dr Steve Lester at [Email Address Removed].
The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.
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