Singmaster's conjecture in the interior of Pascal's triangle

In 1971, David Singmaster conjectured that any natural number greater than one only appears in Pascal's triangle a bounded number of times. In the talk I will discuss what is known about this conjecture, concentrating on a recent result in joint work with Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen that establishes the conjecture in the interior region of the triangle. An important analytic input in our proof is Vinogradov's estimate for exponential sums over primes.