Details of Grant 

EPSRC Reference:	EP/M025365/1
Title:	Analytic methods for large discrete structures
Principal Investigator:	Kral, Professor D
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	University of Warwick
Scheme:	Standard Research
Starts:	01 October 2015	Ends:	30 September 2018	Value (£):	301,248
EPSRC Research Topic Classifications:	Logic & Combinatorics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
The proposed research belongs to discrete mathematics. The project addresses several fundamental questions in the recently emerged and rapidly evolving theory of combinatorial limits. This theory led to a better understanding of important concepts in combinatorics and theoretical computer science, e.g. regularity and property testing, it provided analytic tools that were used to solve many long standing open problems in extremal combinatorics, and it opened new links between analysis, combinatorics, ergodic theory, group theory and probability theory. One of the reasons for the rapid growth of the theory of combinatorial limits comes from computer science where structures such as the graph of internet connections or graphs of social networks (e.g. Facebook, LinkedIn) are of enormous size and the tools from theory of combinatorial limits can be used to approximate these structures by analytic objects. We plan to extend the range of possible applications of the flag algebra method in extremal combinatorics by adopting the method to new settings, in particular those related to Turán densities, and give applications of the method to specific problems from extremal combinatorics. We will also use the flag algebra arguments to gain new insights in the relation between finitely forcible graph limits and optimal solutions of extremal combinatorics problems. The concepts related to the limits of dense and sparse discrete structures were developed to a large extent separately and an attempt to bridge the gap between them was made through the model theory inspired notion of FO convergence. We plan to explore the limits of representing FO convergent sequences by the corresponding analytic objects (modellings). Our intention is also to use the knowledge gained to provide a more robust notion of convergence which would be less sensitive to minor local modifications.
Key Findings
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Potential use in non-academic contexts
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Organisation Website:	http://www.warwick.ac.uk