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Finite and Descriptive Combinatorics

Project description

Exploring connections between combinatorial and analytical structures

The EU-funded FDC project aims to expand the existing connections and build new bridges between various areas of mathematics, focusing on combinatorial structures and analytic objects. Researchers will seek to successfully adapt combinatorial methods to find constructive solutions to analytic problems whose currently known solutions rely on the axiom of choice. Promising connections between descriptive set theory, efficient distributed algorithms and invariant random processes on infinite vertex-transitive graphs will be explored. In addition, researchers will apply the emerging theory of limits of discrete structures to tackle key problems of external combinatorics.

Objective

This project will explore emerging deep connections and build new bridges between some areas that study finite combinatorial structures (such as extremal and probabilistic combinatorics, distributed algorithms, etc) and those that study analytic objects (such as the limit theory of discrete structures, descriptive set theory, measured group theory, random processes on infinite graphs, statistical physics, etc), with applications going both ways.

One part of this project is to apply combinatorial methods in search of constructive answers to analytic problems whose currently known solutions rely on the Axiom of Choice. One such direction is to investigate a possible transference principle that allows to turn some existence results for finite graphs obtained via the very powerful occupancy method into measurable solutions of the corresponding problems of descriptive combinatorics. Similarly, the project will explore promising connections between descriptive set theory, efficient distributed algorithms, invariant random processes on infinite vertex-transitive graphs, etc. Some problems that the project will investigate from this point of view are the Spectral Gap Conjecture, Mycielski's divisibility problem, and the existence of measurable graph factors and colourings.

Also, various important unsolved problems of extremal combinatorics will be approached via the limits of discrete structures (which are analytic objects that encode large-scale properties). In addition to using some established techniques (such as flag algebras and the stability method), the project will look for novel ways of applying the analytic aspects of limit objects that have a great potential in this respect. New software for general-purpose flag algebra calculations will be written and made freely available.

The project will also study some general fundamental questions about graph limits (such as approximability by finite graphs, identification using partial subgraph counts, etc).

Host institution

UNIVERSITY OF WARWICK
Net EU contribution
€ 1 581 387,00
Address
KIRBY CORNER ROAD UNIVERSITY HOUSE
CV4 8UW COVENTRY
United Kingdom

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Region
West Midlands (England) West Midlands Coventry
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 1 581 387,00

Beneficiaries (1)