# Mathematics Colloquia and Seminars

Many conjectures in number theory have analogues for polynomials in one variable over a finite field. In recent work with Mark Shusterman, we proved analogues of two conjectures about prime numbers - the twin primes conjecture and the conjecture that there are infinitely many primes of the form $n^2+1$. The proofs combine elementary, classical identities about the primes with elementary, non-classical manipulations of polynomials and geometric methods. I will give an introduction to these ideas.