Workshop - Thermodynamic Formalism: Ergodic Theory and Geometry
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
The last decade has seen spectacular and continuing advances in an approach to ergodic theory and its applications based on the so-called Thermodynamic Formalism. This approach, which is rooted in statistical physics, has two aspects. On one hand, interesting invariant measures for dynamical systems - known as equilibrium measures and including natural geometric measures - arise as a result of variational principles involving entropy or, to include weighting by a potential, its weighted version topological pressure. On the other, the topological pressure and the associated measures are encoded as eigendata of a family of bounded linear operators, called transfer operators, which enables a wealth of operator-theoretic tools to be employed. In the most favourable circumstances, where the system and potential are analytic, these operators will be nuclear (in the sense of Grothendieck) and so their spectrum consists of eigenvalues and they have a well-defined trace. This, in particular, allows the definition of the associated zeta function as a meromorphic function on the complex plane and the accurate computation of many numerical characteristics. More generally, considerable effort has gone into finding new function spaces that facilitate the analysis of a range of systems.
This theory is best developed in the hyperbolic setting and the basic framework was put in place by Bowen, Ruelle and Sinai by the 1970s. A major advance was given by Dolgopyat's work in the 1990s, which allowed estimates on transfer operators as the potential varies, which he used to prove exponential mixing rates for a wide class of hyperbolic flows. We are now seeing a new wave of development of this theory, with new advances in the hyperbolic setting and extensions to situations beyond uniform hyperbolicity, such as systems with singularities (such as the Lorenz attractor) and non-uniformly hyperbolic systems, and to open systems. This, in turn, has opened up new geometric applications, for example to rank 1 non-positively curved manifolds and to spaces exhibiting coarse negative curvature.
The aim of the proposal is to hold an intensive research workshop to progress the use of Thermodynamic Formalism as a method in ergodic theory and applications. Such a workshop is timely as the subject has seen rapid development in the last few years, with both new applications to other areas and fresh injections of ideas coming from other branches of analysis. Examples of the former are the use of thermodynamic ideas to carry out rigorous computations of dimensions of fractal sets and other numerical characteristics, and the application of ideas from work on the decay of correlations for Lorenz systems to the problem of characterising mixing for the Weil-Petersson geodesic flow in geometry. Examples of the latter are the use of the so-called `fractal uncertainty principle', developed as part of the theory of mathematical analysis on fractals, to study essential spectral gaps for open dynamical systems, and the use of the intrinsic negatively curved geometry present in some systems to employ thermodynamic methods without the requirement of constructing a symbolic model, which is only available in more restricted settings. The objectives are to attack a number of specific problems with progress being measured in terms of both solutions and partial solutions, and the introduction of new ideas.
This theory is best developed in the hyperbolic setting and the basic framework was put in place by Bowen, Ruelle and Sinai by the 1970s. A major advance was given by Dolgopyat's work in the 1990s, which allowed estimates on transfer operators as the potential varies, which he used to prove exponential mixing rates for a wide class of hyperbolic flows. We are now seeing a new wave of development of this theory, with new advances in the hyperbolic setting and extensions to situations beyond uniform hyperbolicity, such as systems with singularities (such as the Lorenz attractor) and non-uniformly hyperbolic systems, and to open systems. This, in turn, has opened up new geometric applications, for example to rank 1 non-positively curved manifolds and to spaces exhibiting coarse negative curvature.
The aim of the proposal is to hold an intensive research workshop to progress the use of Thermodynamic Formalism as a method in ergodic theory and applications. Such a workshop is timely as the subject has seen rapid development in the last few years, with both new applications to other areas and fresh injections of ideas coming from other branches of analysis. Examples of the former are the use of thermodynamic ideas to carry out rigorous computations of dimensions of fractal sets and other numerical characteristics, and the application of ideas from work on the decay of correlations for Lorenz systems to the problem of characterising mixing for the Weil-Petersson geodesic flow in geometry. Examples of the latter are the use of the so-called `fractal uncertainty principle', developed as part of the theory of mathematical analysis on fractals, to study essential spectral gaps for open dynamical systems, and the use of the intrinsic negatively curved geometry present in some systems to employ thermodynamic methods without the requirement of constructing a symbolic model, which is only available in more restricted settings. The objectives are to attack a number of specific problems with progress being measured in terms of both solutions and partial solutions, and the introduction of new ideas.
Planned Impact
The proposed workshop will bring together a group of world-leading experts on topics revolving around Thermodynamic Formalism.
The themes of the workshop are rooted in the mathematical sciences (ergodic theory, dynamical systems), with applications to a range of mathematical areas and beyond (geometry, geometric group theory, biology, physics, neuroscience), addressing some key problems with the capacity for long-term impact on the UK research base in the Mathematical Sciences.
At the level of individual researchers, this workshop has the potential to represent a career-changing interaction - this is highly relevant to the EPSRC goal of delivering highly skilled numerate people, with the absorptive capacity to use knowledge and technologies which may have been developed elsewhere; in particular, a number of UK PhD students will participate in the workshop.
Society:
A very novel and topical direction of recent research in ergodic theory is its application to DNA sequencing and neural complexity. D. Thompson (Ohio State), will be chairing special workshop sessions on applications of thermodynamic formalism, with particular emphasis on making progress in its applications to mathematical biology. Advances stemming from these organised sessions, together with associated breakout activities, will be of clear relevance to biologists, neuroscientists, and other applied scientists working in academia and related industries. Ultimately, progress on DNA sequencing and neural complexity represent tangible benefits to wider society, and fall within the Healthy Nation Prosperity Outcome of the EPSRC Delivery Plan.
Economy:
Another key theme of the workshop concerns algorithms for computing and approximating physically relevant quantities. This topic has the potential to impact, in the short to medium term, on a wide range of academic applications in information theory and the physical sciences, including Lyapunov exponents (important for the stability of solutions of equations), entropy rates (for hidden Markov processes and binary symmetric channels in information theory), and linear response (important in meteorology and earth sciences). More broadly, algorithms are central to the future of the UK, and currently used across a range of industries (e.g. data analytics of big data, meteorology, medical imaging (CAT scans)), so that progress in the refinement of dynamical algorithms emanating from the workshop could, in the medium to long term, impact upon the wider economy. Specifically, the potential application to business innovation via digital transformation falls within the Productive Nation Prosperity Outcome of the EPSRC Delivery Plan, while the development of algorithms (and potential connections with the Data-driven Economy and the Internet Of Things) is highly relevant to the Connected Nation Prosperity Outcome.
The themes of the workshop are rooted in the mathematical sciences (ergodic theory, dynamical systems), with applications to a range of mathematical areas and beyond (geometry, geometric group theory, biology, physics, neuroscience), addressing some key problems with the capacity for long-term impact on the UK research base in the Mathematical Sciences.
At the level of individual researchers, this workshop has the potential to represent a career-changing interaction - this is highly relevant to the EPSRC goal of delivering highly skilled numerate people, with the absorptive capacity to use knowledge and technologies which may have been developed elsewhere; in particular, a number of UK PhD students will participate in the workshop.
Society:
A very novel and topical direction of recent research in ergodic theory is its application to DNA sequencing and neural complexity. D. Thompson (Ohio State), will be chairing special workshop sessions on applications of thermodynamic formalism, with particular emphasis on making progress in its applications to mathematical biology. Advances stemming from these organised sessions, together with associated breakout activities, will be of clear relevance to biologists, neuroscientists, and other applied scientists working in academia and related industries. Ultimately, progress on DNA sequencing and neural complexity represent tangible benefits to wider society, and fall within the Healthy Nation Prosperity Outcome of the EPSRC Delivery Plan.
Economy:
Another key theme of the workshop concerns algorithms for computing and approximating physically relevant quantities. This topic has the potential to impact, in the short to medium term, on a wide range of academic applications in information theory and the physical sciences, including Lyapunov exponents (important for the stability of solutions of equations), entropy rates (for hidden Markov processes and binary symmetric channels in information theory), and linear response (important in meteorology and earth sciences). More broadly, algorithms are central to the future of the UK, and currently used across a range of industries (e.g. data analytics of big data, meteorology, medical imaging (CAT scans)), so that progress in the refinement of dynamical algorithms emanating from the workshop could, in the medium to long term, impact upon the wider economy. Specifically, the potential application to business innovation via digital transformation falls within the Productive Nation Prosperity Outcome of the EPSRC Delivery Plan, while the development of algorithms (and potential connections with the Data-driven Economy and the Internet Of Things) is highly relevant to the Connected Nation Prosperity Outcome.