FANTAST · Frontiers of Analytic Number Theory And Selected Topics

This proposal sits at the interface of analytic number theory and selected topics, viewed through the prism of Diophantine equations defining higher-dimensional algebraic varieties. A core part of the proposal involves using analytic methods (such as complex analysis, Fourier analysis and additive combinatorics) to tackle a range of problems about Diophantine equations. These include suchbasic questions as precisely when families of equations admit integer or rational solutions and, furthermore, how ``dense'' these solutions are when they exist. In the reverse direction, a significant component of the proposal is dedicated to established problems in number theory (such as stable cohomology of moduli spaces and uniform spectral gaps for arithmetic lattices) which can be tackled via the successful analysis of intermediary Diophantine equations.""