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Geometric Measure Theory and Fractal Geometry
Fundamental geometric and dimensional properties of sets and measures
remain a major interest. Continuing work, partly in collaboration with
mathematicians from other universities, includes the geometry of
Hausdorff and packing measures, including sections and projections of
fractals and the analysis of self-affine sets and measures.
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Dynamics and Ergodic Theory
Although the ergodic theorem guarantees that ergodic averages converge to
an almost sure limit, the sets where the upper and lower limits take
specific different values can also be large. Olsen has introduced powerful
techniques allowing analysis of such sets of 'divergence points'.
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Multifractal Geometry
The measure theoretic multifractal formalism introduced in the 1990s by
Olsen is now standard in rigourous multifractal analysis and has been used
in many questions involving analysis of measures. Topics studied from a
multifractal viewpoint include self-affine measures, divergence points, points of non-differentiability of
functions and inhomogeneous measures.
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Metric Number theory
The Group has worked on many aspects of metrical Diophantine
approximation, the structure of the Liouville numbers and metric
properties of continued fractions. Olsen and collaborators have studied
topological and metric properties of non-normal numbers.
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Stochastic processes
Gaussian and stable processes are used increasingly to model highly
irregular phenomena. Work here includes the study of local properties of
such processes, which has demonstrated the ubiquity of the fractional and
multifractional Brownian motions. Collaborations on fractal stochastic processes with
French institutions have let to the
introduction of 'multistable' processes which promise to be widely
applicable.
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Noncommutative Fractal Geometry
Current work is developing
spectral triples for fractal spaces and on noncommutative versions of the
thermodynamic formalism.
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Group Theory and Fractals
A novel group-theoretic approach was introduced (jointly with O'Connor
from the Algebra Group) to analyse the symmetry structure of classes of
fractal. The fractal geometry of subgroups of the automorphism groups of
infinite trees is also starting to be fruitful.
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Differential equations on fractals
A program started with Jiaxin Hu studies nonlinear diffusion
processes and PDEs on fractal domains, with an emphasis on the interplay
between the various dimensions of the domain and the critical parameters
for the behaviour of solutions.
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