EPSRC Reference: 
EP/M001903/1 
Title: 
Applications of ergodic theory to geometry: Dynamical Zeta Functions and their applications 
Principal Investigator: 
Pollicott, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
EPSRC Fellowship 
Starts: 
30 December 2014 
Ends: 
29 December 2019 
Value (£): 
934,489

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Physics 
Numerical Analysis 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The proposed research lies at the interface of Ergodic Theory and Dynamical Systems, geometry, number theory, partial differential operators and mathematical physics. Central to this research programme are the the application of ideas from smooth ergodic theory to problems in different areas of mathematics. As such it is a highly intradisciplinary research program. It also seems very timely, since there has been an explosion of activity in these areas in the last year which has attracted widespread attention. The proposed research is at the cutting edge of this development. In particular, the basis for this project rests on four important interrelated strands in applications of ergodic theory and dynamical systems to other areas: zeta functions and Poincare series (with their connections to number theory and geometry); Decay of correlations and resonances (with applications to the physical sciences); Numerical algorithms (with applications to both Pure and Applied Mathematics); and Teichmuller theory and WeilPetersson metrics (at the boundary of ergodic theory, analysis and geometry).
The study of geometric zeta functions for closed geodesics on negatively curved manifolds was initiated by Fields Medallist A. Selberg in the 1950s (following his earlier work on number theory). Selberg studied the case of constant curvature manifolds, using trace formulae and ideas from representation theory which do not generalise. However, recent work of Giulietti, Liverani and myself used a completely different viewpoint involving ideas in ergodic theory to extend the zeta function for negatively curved manifolds (and even more generally smooth Anosov flows, generalizing the geodesic flow). This provides the starting point for our proposed research on zeta functions, providing both a springboard to a whole host of significant applications and providing the scientific framework via the new ideas and techniques it initiated.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 