The Quadratic Littlewood-Offord Problem

Explore the quadratic Littlewood-Offord problem through this 58-minute mathematical lecture that examines how quadratic polynomials with random binary inputs can concentrate on single values. Delve into the fundamental question of concentration for quadratic polynomials Q(x₁,...,xₙ) when evaluated at independent unbiased {1,-1}-valued random variables, extending the classical Littlewood-Offord problem from linear to quadratic cases. Learn about the problem's significance in the study of symmetric random matrices as popularized by Costello, Tao, and Vu, and discover the essentially optimal bound obtained through joint work with Matthew Kwan that confirms the conjecture by Nguyen and Vu. Gain insights into advanced probability theory, random matrix theory, and combinatorial mathematics while exploring connections to o-minimal structures and the broader conjecture by Fox, Kwan, and Spink for subsets of ℝᵈ.