We start with an image varying from white at the left to black at the right.
The image at the top right shows what happens to this image when the
standard map
is applied for 20 iterations. This map is a "chaotic map" which is an
abstraction of many dynamical systems. It is mixing in some regions, but is not
mixing
in the ergodic theory sense.
The bottom image shows the magnitude of the Haar wavelet basis coefficients
of the top image. Wavelets
are a generalisation of the Fourier transform: typically one looks for expansions
of a suitable Hilbert space
of functions, which reflect both spatial and frequency data. The classical Fourier
transform only captures frequency data: for this image, we would lose interesting
information about the spatial mixing.
Within the Analysis group, Dr Kisil and Prof Partington study wavelets.
Prof Partington is interested in the abstract Hilbert space theory of wavelets,
as well as applications to signal processing and control theory. Dr Kisil is
interested in applications to Mathematical Physics.
Ergodic theory in applied mathematics typically studies the iterations of a single
map, or a oneparameter (semi)group of maps. In pure mathematics, it is interesting
to generalise this to the actions of general groups on spaces. This leads to an
incredibly powerful way of forming interesting topological algebras: the
crossed product construction
in von Neumann algebras.
We see that a single image has links to much work done at Leeds! Functional
analysis is a subject which sits in the middle of modern mathematics, being informed
by many different areas, and having diverse applications both in mathematics, and
in wider fields.
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