The concepts of Randomness and Expansion are pervasive throughout Mathematics and its applica- tions to many areas of Science and Engineering. The mathematical study of Expansion can be traced back to the ancient Greeks and of Probability to the analysis (e.g. by Fermat and Pascal) of games of chance. In the modern era, both concepts are influential in many areas of Mathematics (this proposal will emphasise Combinatorics and Probability, and also touch on Analysis, Geometry, Topology, Number Theory and Theoretical Computer Science). Within Science and Engineering, topics related to the mathematical problems covered in this proposal include Approximation Algorithms (Counting and Sampling), Statistical Physics (Magnetism, Lattice Gases, Polymer Models), Mathematical Biology (Epidemiology), Control Theory and Fluid Flow.
My recent and ongoing research has generated several exciting new ideas and methods. The most recent of these, the Cluster Expansion Method (work with Jenssen), is a far-reaching program to apply a classical tool from Statistical Physics to my objective of developing methods for describing the typical structure of models such as random homomorphisms from a discrete torus. Another exciting recent technique, Global Hypercontractivity (work with Lifshitz, Long and Minzer), is a structural refinement of the classical hypercontractivity theorem; I will generalise many of its applications to Mathematics and Computer Science and give several new applications, e.g. in Extremal Combinatorics (via the Junta Method). I will also develop new Absorption tech- niques to answer constructive mathematical questions that seem beyond the reach of Randomised Algebraic Construction (a method I developed to solve Steiner’s 1852 question on the Existence of Designs) such as the existence of Steiner Triple Systems of high girth or bounded degree high-dimensional expanders.
