| Description |
Diophantine equations and prime numbers form two important
research areas in number theory, and it is therefore natural to
consider a hybrid problem from these two areas: the existence of prime
solutions to Diophantine equations. In this talk I will report a joint
result with Peter Sarnak in this direction, on almost-prime solutions
to quadratic equations in three variables. Our strategy differs from
the previous ones by applying ideas from various fields of mathematics
such as the algebraic theory of quadratic forms, the Jacquet-Langlands
correspondence in the theory of automorphic representations, the
harmonic analysis method of Selberg, as well as nonlinear
combinatorial sieves.
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