 | |
 |
|
 |
|
| Mathematical Physics comprises in part the
work on Integrable Systems as well
as in
general the more mathematical aspects
of
physical systems, for example quantum
systems.
This may involve quantum field theoretical
models or models in statistical mechanics.
The Applied Analysis work in the department
concentrates on multi-parameter spectral
methods, linear and nonlinear special
functions
and associated quantum (integrable)
systems
both continuous as well as on the space-time
lattice. |
|
 |
| Current trends in microwave semiconductor
technology take advantage of developments
in low-dimensional structures using
hetero-junctions.
An example of this is the High Electron
Mobility
Transistor (HEMT) using Gallium Arsenide
and Aluminium Gallium Arsenide in which
electrons
suffer little scattering. Manufacturers
of
these devices are constantly trying
out different
layer structures in order to improve
performance,
and require fast and accurate solutions
of
the modelling equations. We are looking
at
this HEMT problem and others in two
main
ways: testing the various assumptions
behind
the equations using relatively simple
numerical
codes, and developing new numerical
codes
for more efficient solutions. |
|
 |
| The theory of integrable discrete systems
also extends into the quantum domain,
and
it is here that the true richness of
integrability
is most visible (quantum mechanics
being
in essence a theory of discrete and
algebraic
objects). The current research concentrates
on formulating a proper quantum theory
for
discrete mappings and integrable systems
living on the space-time lattice. Exploiting
the exactness of the models and integrable
structures (R-matrices, quantum Lax
pairs
and determinants) the investigation
is set
to produce rigorous and analytic answers
to questions which for other models
are only
accessible through numerical and perturbative
methods. As such, quantum integrable
discrete
models form a paradigm for the development
of new approaches in the quantum regime
which
have potential implications to areas
such
as string and conformal field theory,
as
well as in the theory of random matrices
in mesoscopic physics, quantum computing
and nanotechnology. |
|
 |
| Understanding the underlying mechanisms of
many nonlinear mathematical models
arising
in Physics, Biology and Medicine calls
for
the development and investigation of
new
analytical tools. For example many
biological
problems are modeled through reaction-diffusion
systems of quasi-linear parabolic differential
equations. Techniques based on the
maximum
principle and differential inequalities
are
being unified to provide tools with
which
to study qualitative behaviour of many
biological
and medical systems including morphogenesis,
cell dynamics, as well as angiogenesis,
vascularisation
and metastasis of malignant tumours.
In particular
theorems have been obtained which can
be
used to establish the existence and
stability
of traveling wave solutions. This in
turn
has led to a novel numerical method
for their
calculation. Ideas drawn from a study
of
the theory of reinforced random walks
is
helping with the understanding of biological
processes at the molecular and cellular
level. |