EPSRC Reference: 
EP/N025636/1 
Title: 
Positivity problems at the boundary between combinatorics and analysis 
Principal Investigator: 
Sokal, Professor A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
UCL 
Scheme: 
EPSRC Fellowship 
Starts: 
01 September 2016 
Ends: 
31 August 2021 
Value (£): 
836,383

EPSRC Research Topic Classifications: 
Logic & Combinatorics 
Numerical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Combinatorics is the branch of mathematics concerned with counting finite
structures of various types (permutations, graphs, etc.); it has
applications in computer science, statistical physics, molecular biology,
and many other fields. Analysis, by contrast, is the branch of mathematics
concerned with continuous variation (i.e. functions of real or complex
numbers); it has applications in nearly all fields of science and engineering.
The proposed research lies at the interface between combinatorics and
analysis: it involves using combinatorial tools to study analytic problems,
and vice versa. More specifically, the proposed research comprises
three themes, all of which are aimed at exploring novel positivity properties
that arise at the interface between combinatorics and analysis.
The first theme involves studying situations in which inverse powers
of combinatorially important polynomials have Taylor expansions with
positive coefficients. The second theme involves studying situations
in which certain matrices formed from sequences of combinatorially
important polynomials have a property called "total positivity".
The third theme involves studying situations in which certain power
series formed from combinatorially important polynomials (for example,
the counting polynomials of connected graphs) have positive coefficients.
This latter property was discovered empirically by the PI in many
situations, but most of these have not yet been proven, and their deeper
meaning remains to be elucidated.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


Sectors submitted by the Researcher 
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Project URL: 

Further Information: 

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