Analysis
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Most of the research in analysis conducted here at King's is related to the spectral theory of linear operators. This can be viewed as an infinite-dimensional generalisation of the eigenvalue theory of nxn matrices. A large part is devoted to the study of those aspects of spectral theory which are relevant to the study of partial differential operators. We are principally concerned with problems of a pure mathematical character, using techniques from Fourier analysis, Lp theory, quantum theory, pseudodifferential operators on manifolds and geometrical matters connected with the wave equation. All of this is heavily functional analytic, and indeed one of us is interested only in functional analytic problems involving non-self-adjoint operators, invariant subspaces and nest algebras. The analysis group run a research seminar jointly with Imperial College. This meets every two weeks and has two invited speakers at every meeting. In addition to this there is an irregular internal seminar in which research students and other members of staff present their recent results for discussion. |
| Number Theory | The research interests of the number theory group
focus on the representation theory and harmonic analysis of p-adic
groups and its arithmetical applications in algebraic geometry and Galois
module theory, and on problems concerning the existence and basic properties
of arithmetical p-adic L-functions.
The work done in King's on aspects of p-adic groups is becoming increasingly
regarded as essential to making further progress with the (local) Langland's
program. This is one of the most fundamental and far reaching areas of
research in pure mathematics today.
Research in the area of arithmetical algebraic geometry focuses on studying
the interaction between structural invariants attached to motives defined
over number fields and the analytic properties of the associated L-functions.
This is a very active area of research and is producing considerable new
insight into long standing problems in the field of Galois module theory.
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| Geometry and Topology | The interests of the geometry and topology group
include the geometry
of Riemann surfaces and discrete groups, as well as their associated moduli
spaces. The K-theory of the modular group is also a current focus of attention.
Several of the topics studied by this group are closely related to mathematical
physics: for example, the study of super Riemann surfaces, which is related
to string theory, and that of Witten invariants of three and four dimensional
manifolds, which is related to objects called quantum groups, whose origins
lie in statistical mechanics.
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| Theoretical Physics | The research interests of the theoretical
physics group are in supersymmetry, string theory and in the construction
of unified theories of the fundamental forces of nature. Current work is
centred around two-dimensional quantum field theories and their applications
to string theory. These field theories have many remarkable properties.
In particular, their symmetries involve a variety of extremely interesting
algebraic structures, and the study of these and their implications is
an active area of research at King's.
Another development coming from the study of two-dimensional field theories
is an extension of the notion of a group, and associated with this idea
is a generalisation of geometry called non-commutative geometry. One of
the best-developed areas of non-commutative geometry is supergeometry,
which is useful in the study of supersymmetric theories.
The theory of Feynman path integrals in such theories is a further active
area of research here at King's.
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| Statistical Mechanics &
Quantum Probability |
Research work in equilibrium statistical mechanics
includes the use of renormalization group and transfer matrix methods to
investigate phase transitions in lattice systems and
the use of scaling methods to investigate many-magnon systems.
In non-equilibrium statistical mechanics, there is interest in the derivation
of the irreversible equations for macroscopic systems with reversible underlying
dynamics; disorded systems; non-equilibrium thermodynamics, applications
of statistical dynamics and that part of the theory of non-linear dynamical
systems that can be formulated in terms of probability theory, either classical
or quantum.
Further interests include the mathematical analysis of interrelationships
between the operator constructs of quantum field theory and classical probability
and the development of non-commutative extensions.
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| Neural Information Processing |
Due to their complex structure, the large number of interacting
elements involved,
and their dynamic nature, networks of interconnected (natural or artificial)
information processing units (neurons) exhibit a highly non-trivial
and very rich behaviour, posing many new
fundamental and challenging mathematical problems.
The main research interests in this group concentrate on the modelling
and analysis of the stochastic dynamics of operation and learning in natural
and synthetic neural networks. There are strong links with statistical
mechanics (stochastic processes, disordered systems, replica theory, functional
integrals), biology (neurophysiological modelling), and computer science
(information theory). See also the Centre for Neural Networks |
| Space Science | Current research involves investigation of the
propagation of whistler-mode waves in the Earth's ionosphere and magnetosphere.
These plasma waves, whose frequencies lie below the electron gyrofrequency,
are guided by enhancements of plasma density (ducts) aligned with the geomagnetic
field, and so may propagate through the magnetosphere from the northern
hemisphere to the southern hemisphere, or vice versa. The waves are strongly
dispersive, and their observation has provided a great deal of information
about the magnetosphere. Propagation in ducts is analogous to that of light
in optical waveguides, though the anisotropy of magnetized plasma introduces
extra complications. Problems being studied include excitation of, and
radiation from ducts, and the effects of fine structure.
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| Theory of Relativity | Currently research in this area is concentrated
on topics in classical general relativity. The general aim of the research
is to obtain a better understanding of the geometrical structure of Einstein's
theory and to develop formalisms which will be useful in either classical
or quantum gravity. Examples of areas of recent interest and activity include
the study of complex structures and Einstein's gravitational field equations,
an extension of the chiral Lagrangian formulation of general relativity
to a new unified approach to Einstein-Yang-Mills theory and the development
of a new variables canonical formalism for gravity which is based on null
hypersurfaces.
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| Mathematical Biology | This field of research includes (i) the investigation
of the general mathematical properties of models of development and their
implications for evolution, and (ii) control in physiology and geophysiology.
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| Mathematical Education | The main area of interest at King's concerns the capacity of the short-term working memory and the extent that this can be effectively `increased' in mathematical thinking, a general aim being to gain an understanding of, and find ways to promote, the phenomenon in which a collection of related items - processes, sentences, representations, objects, properties, or steps of logical deduction - become mentally compressed into one single mental entity. Two recent studies arising from this concern the nature of students' difficulties in negating statements involving quantifiers, and first year students' responses to cues in proving the irrationality of the square roots of 2 and 3. Currently research is concerned with the development of algebraic ability in school pupils at Key Stage 4.
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