The objective of this research project is to develop new and original methods to address a number of fundamental questions in the development of smooth ergodic theory and dynamical systems. Furthermore, this theory will have important and diverse applications to many different areas of mathematics (in particular geometry, but also number theory and topology).
The questions we are interested in are specifically related to the study of resonances, which characterise statistical properties of dynamical systems via the correlation function and the zeta function.
The generality of the approach draws heavily on the PI's experience with the ergodic theory of hyperbolic systems, an area of research in which the PI has played a leading role for many years, but the risk and ambition stems from the novelty and risk inherent in trying to apply new ideas.
There are three major strands to this work although they are bound together by interactions both at the level of the conclusions and the methodology.
(a) The first is the development of new methods for determining numerically basic characteristic values, which are theoretically poorly understood, for classical (``chaotic'') hyperbolic systems. The important new development is a completely rigorous estimate on the approximation.
(b) The second main theme will be the establishment of a radically new theory of resonances and correlation functions for a broader classes of systems, for which there is presently no existing theory. Typically these examples arise in geometry, and will help advancing our understanding of problems in that area.
(c) The third main topic will be applications to specific major problems, particularly in hyperbolic geometry, and number theory.
A key ingredient in our approach is the blending of classical ideas from classical thermodynamic formalism with more progressive functional analytic techniques that continue to emerge.
Field of science
- /natural sciences/mathematics/pure mathematics/arithmetic
- /natural sciences/mathematics/pure mathematics/geometry
- /natural sciences/mathematics/applied mathematics/dynamical systems
Call for proposal
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