EPSRC Reference: 
EP/K000799/1 
Title: 
A transfer operator approach to Maass cusp forms and the Selberg zeta function 
Principal Investigator: 
Pollicott, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
Standard Research 
Starts: 
04 February 2013 
Ends: 
03 February 2016 
Value (£): 
268,076

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
04 Jul 2012

Mathematics Prioritisation Panel Meeting July 2012

Announced


Summary on Grant Application Form 
Zeta functions have played a central role in mathematics for many centuries and the most famous example, the Riemann zeta function, plays a pivotal role in analytic number theory and the proof of the famous Prime Number Theorem, and is also the subject of the famous Riemann Hypothesis. The study of zeta functions in number theory and related fields lead very naturally to the introduction, in 1956, of the Selberg zeta function. In particular, this is a function of a single complex variable which is analogous to the Riemann zeta function, but is defined in terms of characteristics of the geometry of Riemann surfaces rather than in terms of prime numbers.
One of the principle advantages of the Selberg zeta function over the Riemann zeta function is the more explicit characterization of its zeros. There is a classical approach to the Selberg zeta function using the LaplaceBeltrami operator, which is a second order differential operator whose eigenvalues describe the zeros of the Selberg zeta function viewed as a complex function. In particular, this is a setting where the HilbertPolya operator theoretic approach to the analogue of the Riemann hypothesis has been successful.
On the other hand, there are certain aspects of the study of the Selberg zeta function where the successes of the spectral method can be augmented by a different approach. More precisely, there is a more modern dynamical approach originating in the pioneering work of Ruelle which brings together ingredients from mathematical physics, operator theory and ergodic theory. In the past decade, there has been considerable interest in the interplay between the classical and modern approaches to understanding the Selberg zeta function and its connections with number theory.
Of particular interest is the use of the dynamical approach to prove results which appear inaccessible by more classical methods. For example, a particular theme of this proposal is the definition and investigation of period functions, which are intimately related to the spectrum of the LaplaceBeltrami operator. Whereas these important functions are fairly intractable using existing techniques, but the aim of this proposal is to analyse them by developing new approaches along the lines. We have every hope that this work will contribute significantly to the development of this important area of mathematics.

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