Terence Tao: Singmaster's conjecture in the interior of Pascal's triangle

An old conjecture of Singmaster asserts that every integer greater than 1 occurs only a bounded number of times in Pascal's triangle. In this talk we survey some results on this conjecture, and present a recent result in joint work with Kaisa Matomaki, Maksym Radziwill, Xuancheng Shao, Joni Teravainen that establishes the conjecture in the interior region of the triangle. Our proof methods combine an "Archimedean" argument due to Kane (and reminiscent of the Bombieri-Pila determinant method) with a "non-Archimedean argument" based on Vinogradov's exponential sum estimates over primes.