Prof/Deputy Head of Department
4 West 4.12
Dept of Mathematical Sciences
Anthony Dooley has worked across a broad area of mathematical analysis. Analysis is that branch of mathematics which deals with continuous phenomena, and while its origins are in the study of physical phenomena, its current applications are widespread and include such diverse fields as economic and financial modelling, climate and weather prediction, computer networks, not to mention quantum mechanics and relativity theory.
His interests in the theory of Lie groups spring from a fascination with the study of quantum mechanics, and how the geometry of the universe can be used to predict the particles which can exist. The groups which describe the symmetries of these geometries were studied by Lie in the late 19th century, and by now form a major part of modern mathematics. Analysis of these groups can be seen as a far-reaching generalisation of Fourier analysis and Dooley has extended familiar results to this wider setting, involving approximation theory, reconstruction methods, differential operators, random processes, and representation theory.
Dooley also works in the area of dynamical systems, which evolved from Statistical Mechanics to become a mathematical study of how systems evolve as time elapses. The area known as ergodic theory (from the Greek ergos, meaning work) has grown through the development of such notions as entropy and orbit equivalence, and now can be used to model the development of chaos, or progression to some kind of predictable limit in a variety of applications. Dooley has made significant progress in the understanding of non-singular dynamical systems, those where the size (or measure) of a set may vary with time. He has also worked on the study of systems of completely positive entropy, which display the most chaotic behaviour.
Applebaum, D. and Dooley, A., 2015. A generalised Gangolli-Levy-Khintchine formula for infinitely divisible measures and Levy processes on semi-simple Lie groups and symmetric spaces. Annales De L Institut Henri Poincare-Probabilites Et Statistiques, 51 (2), pp. 599-619.
Dooley, A.H., Golodets, V.Y. and Zhang, G., 2014. Sub-additive ergodic theorems for countable amenable groups. Journal of Functional Analysis, 267 (5), pp. 1291-1320.
Dooley, A. H. and Hagihara, R., 2012. Computing the critical dimensions of Bratteli–Vershik systems with multiple edges. Ergodic Theory and Dynamical Systems, 32 (01), pp. 103-117.
Dooley, A. H., Hawkins, J. and Ralston, D., 2011. Families of type III0 ergodic transformations in distinct orbit equivalent classes. Monatshefte fur Mathematik, 164 (4), pp. 369-381.
Dooley, A. H., 2011. Intertwining operators, the Cayley transform, and the contraction of K to NM. Contemporary Mathematics, 544.
Craddock, M. J. and Dooley, A. H., 2010. On the equivalence of Lie symmetries and group representations. Journal of Differential Equations, 249 (3), pp. 621-653.
Pollett, P. K., Dooley, A. H. and Ross, J. V., 2010. Modelling population processes with random initial conditions. Mathematical Biosciences, 223 (2), pp. 142-150.
Danilenko, A. I. and Dooley, A. H., 2010. Simple ℤ2-actions twisted by aperiodic automorphisms. Israel Journal of Mathematics, 175 (1), pp. 285-299.
Dooley, A. H. and Golodets, V. Y., 2009. The geometric dimension of an equivalence relation and finite extensions of countable groups. Ergodic Theory and Dynamical Systems, 29 (06), pp. 1789-1814.
Dooley, A. H., Golodets, V. Y., Rudolph, D. J. and Sinel’shchikov, S. D., 2008. Non-Bernoulli systems with completely positive entropy. Ergodic Theory and Dynamical Systems, 28 (01), pp. 87-124.