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Institute of Advanced Study

Dr Richard C Miles

COFUND Senior Research Fellow at St John's College (April - June 2016)

Dr Richard Miles is a Senior Lecturer in Mathematics at Uppsala University, Sweden

Counting for group actions: structure and dynamics. Dynamical systems – abstract models of physical systems that evolve in time according to fixed physical laws – can be studied using mathematical tools. Abstractly the fixed physical laws become a symmetry reflected by an action of a group (in the case of evolution in time, the group of whole numbers reflecting the state of the system at time 1,2,3,...). There is an extensive body of work on dynamical systems with additional symmetries arising from the specific system being studied, which may be thought of as spatial symmetries. In a different direction, there are natural situations where the additional symmetries also permit evolution in (analogues of) time, reflected in the action of a higher-dimensional group. This leads to the research project, which is to develop the theory of group actions from a dynamical point of view, with particular emphasis on counting problems for group actions. This involves understanding a delicate interaction between the nature of the dynamical system and how it moves points around, the internal structure of the acting group which may be very complex, and the analytic nature of families of associated functions built from the counting problems.

A particular benefit of this Fellowship is the opportunity to work with Professor Tom Ward, with whom Prof Miles has successfully collaborated in the past. Professor Ward was one of the initiators of the study of periodic orbits for higher dimensional actions in the late 1980s and has been a leading exponent since then. There is undoubted scope for them to work together on these problems and this would certainly be of considerable benefit to the research of both.