The exact $k$-SAT threshold for large $k$
We establish the random $k$-SAT threshold conjecture for all $k$ exceeding an absolute constant $k_0$. That is, there is a single critical value $\alpha_*(k)$ such that a random $k$-SAT formula at clause-to-variable ratio $\alpha$ is with high probability satisfiable for $\alpha$ less than $\alpha_*(k)$, and unsatisfiable for $\alpha$ greater than $\alpha_*(k)$. The threshold $\alpha_*(k)$ matches the explicit prediction derived by statistical physicists on the basis of the one-step replica symmetry breaking (1RSB) heuristic. In the talk I will describe the main obstacles in computing the threshold, and explain how they are overcome in our proof. Joint work with Jian Ding and Allan Sly.