Distance problems in discrete geometry include fascinating examples of questions that are easy to state and hard to solve. Three of the best known problems of this type, raised in the 40s, are the Erd\H{o}s Unit Distance Problem, his Distinct Distances Problem, and the Hadwiger-Nelson Problem about the chromatic number of the unit distance graph in the plane. I will describe surprisingly tight recent solutions of the analogs of all three problems for typical norms, settling, in a strong form, questions and conjectures of Matou\v{s}ek, of Brass, of Brass, Moser and Pach, and of Chilakamarri. I will also discuss a related work about ordering points according to the sum of their distances from chosen vantage points. The proofs combine Combinatorial, Geometric and Probabilistic methods with tools from Linear Algebra, Topology, and Algebraic Geometry. Based on recent joint works with Matija Buci\'c and Lisa Sauermann, and with Colin Defant, Noah Kravitz and Daniel Zhu.
Department of Mathematics