Research (Mathematics)

Dr Christopher Poweles

Dr Powles’s research interests lie mainly in the field of mathematical acoustics. To date, his work has involved the application of analytical techniques to problems related to aeroacoustics (specifically to the noise problems of modern aeroengines), and to problems in underwater acoustics and in loudspeaker design.

SILENCER (DTI)

  • Propagation of fan tones from the bypass duct using the extended Munt method (Wiener-Hopf solution).

TURNEX (EC)

  • Propagation of sound through sheared steady mean flows, for applications to jet blockage.
  • Propagation of sound through unsteady (turbulent) sheared flows, for application to the prediction of spectral broadening of turbine tones.

HISAC (EC)

  • Prediction of jet noise generated in mixer-ejector nozzles, for supersonic business jets.

ANDANTE (DTI)

  • Propagation of sound through sheared steady mean flows, for applications to engine installation effects.

SYMPHONY (TSB)

Scattering of noise by rotating blade rows. Additional work carried out, not linked to any large-scale projects and with funding from a variety of sources including EPSRC, the Nuffield foundation, Rolls-Royce and uclan include

  • The behaviour of energy paths in sound fields.
  • Generation of noise by blade-vortex interaction.
  • Scattering of sound in non-uniform ducts.
  • Noise generation and propagation from open-rotor engines.
  • The use of Green’s function techniques in Computational Aeroacoustics.
  • Remote monitoring of sperm-whale populations.
  • The design of transmission-line loudspeaker cabinets.

Dr Davide Penazzi

Dr Davide Penazzi is interested in Model Theory and Applications. Applied Model Theory is the study of mathematical structures with a viewpoint informed by Logic. In particular, he studies real closed fields (whose axiomatic structure is equivalent to that of the real numbers) with particular attention to Nash groups definable in them; and differential fields (equivalent to the complex numbers equipped with a derivation), especially questions concerning transcendence of solutions of differential equations. He is also interested in topological dynamics: the actions of groups on a compact space, in particular the action of a definable group on the space of its types.

 

Model Theory

Model theory is an active branch of mathematical logic. It has many applications to other areas of pure and applied mathematics. Model theory studies the relationship between mathematical structures and the language we use to describe them. To each mathematical structure a logical language is assigned. This gives us a family of subsets called ‘definable sets’, which we study using tools from both mathematics and logic.

There are two main foci to research in model theory. Pure model theory aims to classify all mathematical structures, creating a ‘geography of mathematics.’ This attempts to show that whole groups of seemingly different mathematical objects share certain basic properties. It has been shown that tame (i.e logically easy to study) structures have a canonical notion of dimension; which allows the development of a geometry in this context. The current drive is to extend this type of result to less tame structures.

Applied model theory looks at the application of this classification to specific structures, giving new insights in many areas of mathematics. The study of o-minimal structures (where there is a specific condition on the definable sets) led to new developments in real geometry and analytic number theory: that of differentially closed fields allows a (new) elegant description of the relationships between solutions of certain differential equations. These are currently very active areas and have attracted non-logicians to model theory.

The UCLAN model theory group is working on both pure and applied model theory. More specifically we are interested in:

  • Topological dynamics
  • Pseudofinite structures
  • Links between category theory and model theory
  • O-minimality
  • Model theory of differential equations.

Model theory is a thriving subject in the UK. The UCLAN group runs a regular joint meeting with Manchester and Leeds (LINK).