Dr. Allen's research lies at the intersection of Homotopy theory and Algebraic Geometry. His current work revolves around certain aspects of Quillen's Conjecture which roughly speaking, seeks to determine if some higher order algebraic structures vanish after a certain degree. His research uses objects called Quasitoric manifolds-mathematical objects that are very rich combinatorially and algebraically--to make progress resolving this conjecture. From an applied perspective, Dr. Allen is interested in Topological Data Analysis and Networking Analysis.