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Combinatorial Applications of Random Processes and Expansion

Project description

Advanced mathematical models to describe real world networks

The concepts of randomness and expansion are pervasive throughout mathematics. The EU-funded CARPE project will build on the success of various exciting new methods based on random process and expansion theories to address challenging problems that concern networks and are inspired by real life phenomena. For example, although there are physical models that can sufficiently describe phase transitions or the spread of information or diseases throughout a network, they cannot be justified mathematically. Researchers will combine various tools to further elucidate the mathematical theories that apply to any possible network. These include a classic statistical physics tool called cluster expansion method as well as global hypercontractivity – a structural refinement of the classical hypercontractivity theorem.

Objective

The concepts of Randomness and Expansion are pervasive throughout Mathematics and its applications 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 Matthew Jenssen), is a far-reaching program to apply a classical tool from Statistical Physics to 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 Noam Lifshitz, Eoin Long and Dor Minzer), is a structural refinement of the classical hypercontractivity theorem; we 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 techniques 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).

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Host institution

THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
Net EU contribution
€ 2 499 010,00
Address
WELLINGTON SQUARE UNIVERSITY OFFICES
OX1 2JD Oxford
United Kingdom

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Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 2 499 010,00

Beneficiaries (1)