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Explore advanced mathematical concepts in this computer science and discrete mathematics seminar that examines high-dimensional expanders (HDX) and their applications through the lens of Lorentzian polynomials. Learn how high-dimensional expanders serve as generalizations of expander graphs with significant applications in coding theory, probabilistically checkable proofs (PCPs), pseudorandomness, derandomization, and approximate sampling. Discover trickle-down theorems as a powerful technique for proving complex structures are HDX, where expansion properties of small components imply expansion characteristics of the entire complex. Examine both established and novel trickle-down theorems specifically designed for approximate sampling applications. Understand how these theorems emerge from the mathematical framework of Lorentzian and log-concave polynomials, which have found diverse applications across mathematics and theoretical computer science. The presentation covers joint research with Kasper Lindberg and Shayan Oveis Gharan, providing insights into cutting-edge developments in high-dimensional expansion theory and its connections to polynomial theory.
