Jonathan Leake, a mathematician at the University of Waterloo, presents this Institute for Advanced Study seminar on high-dimensional expanders, a generalization of expander graphs used in coding theory, probabilistically checkable proofs, pseudorandomness, derandomization, and approximate sampling. Leake works through trickle-down theorems, the technique that lets researchers prove a simplicial complex is a high-dimensional expander by first establishing expansion in small local pieces and then propagating that property up through the whole structure. He surveys both the classical versions of these theorems and new results he develops with Kasper Lindberg and Shayan Oveis Gharan, aimed at applications in approximate sampling algorithms. A central thread is how these trickle-down arguments can be derived from the theory of Lorentzian and log-concave polynomials, a framework that has recently found uses across combinatorics and theoretical computer science. The talk assumes familiarity with expander graphs and simplicial complexes and is pitched at researchers working in discrete mathematics and theoretical computer science.