Explore Terence Tao's lecture on Singmaster's conjecture within Pascal's triangle. Delve into the mathematical intricacies of this old conjecture, which asserts that every integer greater than 1 appears only a bounded number of times in Pascal's triangle. Discover recent results from joint work with Kaisa Matomaki, Maksym Radziwill, Xuancheng Shao, and Joni Teravainen that establish the conjecture in the triangle's interior region. Learn about the proof methods combining Kane's "Archimedean" argument with a "non-Archimedean argument" based on Vinogradov's exponential sum estimates over primes. Examine topics such as binomial collisions, sporadic collisions, upper bounds, controlling bounds for derivatives, counting solutions, privacy intervals, Kumar's theorem, statistical analysis, exponential sums, equidistribution, and covariance structures.