Alternative twin prime problems
Mark Shusterman - Harvard University
It is conjectured that there are infinitely many pairs of primes that differ by 2. After presenting some motivation, results toward the conjecture, and obstructions to further progress, we will consider an analogous problem where the primes are replaced by irreducible polynomials with coefficients in the integers modulo 3 (or modulo 5). Alternative problems of this type often boil down to counting solutions to algebraic equations. Work of Grothendieck and Deligne reduces the latter to (topological) questions about the shape of the geometric figures cut out by our equations. I will report on joint work with Will Sawin obtaining some control on the number of higher-dimensional holes inside the figures in question. This allows us to make progress on some alternative twin prime problems, similar to the ones mentioned above.