Diophantine equations at the interface of analysis and geometry

The purpose of the proposed project is to connect the fields of arithmetic geometry and harmonic analysis via the connections both fields have with counting problems in number theory. The field of arithmetic algebraic geometry seeks to understand the distribution of rational points on varieties.

Analytic methods have long been used in this context, but thanks to progress in harmonic analysis they now give close to optimal results in some cases.

I plan to adapt these results to make them more applicable in geometric contexts, and to explore their effect on such classes of varieties to which these methods are most amenable.Meanwhile, an important question in classical analysis studies the behaviour of the Fourier transform restricted to manifolds.

This Fourier restriction conjecture has a central role within harmonic analysis, and it is closely related to the problem of counting integer solutions of systems of equations.

By employing techniques from arithmetic geometry to these counting problems, I expect to be able to make some conceptual progress beyond what has been attainable by the real-analytic methods that have traditionally been favoured.An important role in both questions is played by the underlying geometry of the problem.

By studying these problems from two different viewpoints, I plan to work towards reconciling the different views both fields have on the role of geometry. I hope that such an improvedunderstanding will enable further progress in both fields.