Traversing Regions of Supersolvable Hyperplane Arrangements and Their Lattice

Explore the mathematical connections between hyperplane arrangements, graph theory, and combinatorial algorithms in this 59-minute conference talk. Discover how to find Hamiltonian paths and cycles in graphs of regions formed by supersolvable hyperplane arrangements, where vertices represent regions and edges connect regions separated by exactly one hyperplane. Learn about the main result proving that supersolvable arrangements always have Hamiltonian cycles in their region graphs, and examine the broader framework of lattice congruences and quotient lattices. Understand how these theoretical findings lead to practical Gray code algorithms for listing various combinatorial objects, including a generalization of the Steinhaus-Johnson-Trotter algorithm for permutations. See applications to well-known supersolvable arrangements that recover existing algorithms while discovering new ones, and explore connections to polytope theory including signed graphic zonotopes and type B quotientopes. Gain insight into the rich interplay between geometry, combinatorics, and algorithmic design through this comprehensive examination of hyperplane arrangement traversal problems.