Download Department of Mathematics Research Colloquia 1998 – 2000 Dec 8 2000

Department of Mathematics Research Colloquia 1998 – 2000
Dec 8 2000 Professor John Stachel
(Boston University)
`New Light on the Einstein-Hilbert Priority Question'
Abstract:
It has been claimed that David Hilbert actually wrote down the field equations of
general relativity shortly before Albert Einstein, and that, on the basis of his mastery
of general covariance, he was better able to understand their physical meaning. On
the basis of recently discovered first proofs of Hilbert's paper on the subject, it is
shown that neither of these claims can be maintained.
Oct 13 2000 Dr. Gordon Woo
(EQE International)
`The mathematics of natural catastrophes'
Abstract:
Natural phenomena have always provided a rich source of problems to challenge
applied mathematicians, and to inspire the creation of new mathematics. Natural
catastrophes are among the most exceptional events in the natural environment, and
mathematicians have an important role in understanding their causes, warning of
their occurrence, forecasting their behaviour, and mitigating their effects. This talk
will review mathematical ideas and concepts which have been applied to natural
catastrophes, and provide practical illustrations from the study of earthquakes,
volcanic eruptions, storms and floods. With over 50,000 fatalities worldwide from
natural catastrophes in 1999, there is no shortage of difficult problems awaiting
urgent solution, and a survey will be presented of some specific issues deserving of
further mathematical attention.
Jun 16 2000 Paul Davies
(Visiting Professor at Imperial College London and Honorary Professor at the University
of Queensland)
`Time travel: Fact or Fiction?'
Abstract:
This was a semi-popular talk without equations - diagrams only - assuming only a
nodding acquaintance with relativity.
Mar 17 2000 A Böttcher (Chemnitz)
"The Spectrum of the Cauchy Singular Integral Operator"
Abstract:
The Cauchy singular integral operator, S, in one of the main actors in the theory of
Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Weiner-Hopf
and singular integral equations. While the boundedness of S has been studied for
many decades, final results on the spectrum of S were obtained only in recent times.
During the last few years, it was discovered that there is a surprising and undreamtof metamorphosis of the (local) spectra of S from circular arcs via horns and
logarithmic double-spirals to so-called logarithmic leaves with a halo. The talk is a
survey of this fascinating development. It is aimed at a general audience with some
inclination to mathematical analysis and concrete operator theory. Technical details
will be omitted, but many beautiful pictures of local spectra will be shown. It will be
a feast for the eyes.
Mar 3 2000 B Derrida (ENS Paris)
"Repetitions in Genealogical Trees and the Renormalization Group".
Abstract:
The number of ancestors of each of us, going backwards in time, doubles at each
generation in the past. This exponential growth makes repetitions unavoidable,
meaning that genealogical trees are not really trees but rather have branches which
coalesce. The statistics of these repetitions obey statistical laws which have simple
scale invariance and can be calculated exactly. This provides a simple example
where Renormalization Group ideas allow to predict non trivial exponents exactly.
In this context, the solution of the Galton Watson process, which describes the
problem of extinction of family names and which is at the origin of the theory of
branching processes can be viewed as a renormalization transformation.
Reference:
B. Derrida, S.C. Manrubia, D.H. Zanette, Universal statistical properties of
genealogical trees, Phys. Rev. Lett. 82 (1999) 1987-1990
Feb 4 2000 L E Fraenkel (Bath)
"On the vertical entry of a wedge into water"
Abstract:
This problem was formulated in 1932 by H. Wagner and has been the subject of
many papers, but seems to have eluded an existence proof until recently. The lecture
will contain an existence theorem and a glimpse of the proof (McLeod and Fraenkel
1997, 2001), but the main intention is to tell stories about the problem and to
describe an aspect of it that remains mysterious.
Jan 21 2000 J Wess (MPI Munich)
Non-commutative Space-time Structures
Abstract:
An old idea to generalize the non-commutative structure of quantum mechanics to
the space coordinates as well has met with new interest recently. Some interesting
examples of a non-commutative space time structure will be discussed in order to
learn what we can expect from such an approach. Most interesting, a lattice-like
structure of space time will arise quite generally.
Dec 10 1999 C Wright (Solipsys Ltd)
Juggling - Theory and Practice
Juggling has fascinated many for centuries. Seemingly oblivious to gravity, the
skilled practitioner can keep several objects in the air at one time, and weave
complex patterns that seem to defy analysis.
In this talk the speaker demonstrates a selection of the patterns and skills of juggling
while at the same time developing a simple method of describing and annotating a
class of juggling patterns. By using elementary mathematics these patterns can be
classified, leading to a simple way to describe those patterns that are known already,
and a technique for discovering new ones.
Nov 26 1999 S Popescu (NI)
What is quantum computation?
"I will present, at an elementary level, the basic principles of quantum computation the main ideas, the architecture of a quantum computer, some quantum
computational algorithms, the problem of noise in quantum computers and proposals
for building quantum computers."
Nov 12 1999 W Hodges (QMW)
The mathematics of language and a question of Dedekind
Oct 29 1999 A Connes (IHES)
Renormalisation in QFT and the Riemann-Hilbert problem
We show that renormalization in quantum field theory is a special instance of a
general mathematical procedure of multiplicative extraction of finite values based on
the Riemann-Hilbert problem.
Oct 15 1999 LMS Meeting in UCL
N Hitchin: "Hyperkähler geometry and integrable systems"
(To be held at University College in the Chemistry Auditorium, Christopher Ingold
Building, Department of Chemistry, 4-5 pm)
Oct 1 1999 P Neumann (Oxford)
What groups were: an account of the development of the axiomatics of group theory
This colloquium will be based on a historical study of axioms for group theory. It
will begin with the emergence of groups in the work of Galois and Cauchy, treat two
lines of development discernible in the latter half of the nineteenth century, and
conclude with a note about some twentieth century ideas. One of those nineteenth
century lines involved Cayley, Dyck and Burnside; the other involved Kronecker,
Weber (very strongly), Hölder and Frobenius.
May 7 1999 Don Zagier (Director, MPI Bonn)
Period Functions for Maass Waveforms
Mar 26 1999 Nick Manton (DAMTP, Cambridge)
Supersymmetric mechanics
Supersymmetric quantum theories are much studied by theoretical physicists. They
are based on supersymmetric classical systems, whose equations of motion are
derived from a Lagrangian. Such Lagrangians, and their accompanying equations,
can seem rather formal. In this talk, I shall present a simple example, the
supersymmetric extension of the mechanics of a particle moving on a line in a
potential. The dynamical variables are Grassmann-algebra valued, with some
commuting and some anti-commuting, but the equations of motion can be explicitly
solved. There are some interesting geometrical features. A large Lie group of
symmetries, related to the abstract supersymmetry algebra, acts on the system.
Mar 12 1999 Steve Alpern (LSE)
Chaos is Generic when Volume is Preserved
Chaotic dynamical systems usually `live' on fairly exotic, often fractal, domains.
However it is possible to show that they can `live' on such ordinary sets as the unit
square. We outline a combinatorial construction which defines an area preserving
chaotic homeomorphism of the square - in particular it gives a homeomorphism of
the square which has a dense orbit. The same idea can be applied to produce chaotic
homeomorphisms on any manifold. Indeed, it shows that for measure preserving
homeomorphisms of a compact manifold, chaos is in fact the general case. This work
is joint with V. Prasad and parallels the work of Aarts, Daalderop, and Fokkink.
Feb 26 1999 June Barrow-Green (Open University)
Poincaré and the three body problem
In 1890 Poincaré's memoir on the three-body problem was published in Acta
Mathematica as the winning entry in the international competition honouring the
60th birthday of Oscar II, King of Sweden and Norway. Today the memoir is
renowned both for providing the foundations for Poincaré's celebrated three volume
Les Methodes Nouvelles de la Mecanique Celeste and for containing the first
mathematical description of chaotic behaviour in a dynamical system. However,
prior to publication Poincaré found a deep and critical error in his original paper and
it was only in correcting the error that he discovered mathematical chaos, with the
result that the published version is very different to the one which he submitted to
the competition. In this talk, I shall discuss Poincaré's paper with particular reference
to the error, while at the same time setting it into a historical context in relation to
the three-body problem itself.
Feb 12 1999 Robbert Dijkgraaf (Amsterdam)
The Unreasonable Effectiveness of Quantum Field Theory in Mathematics
The last decade has seen remarkable applications of ideas from particle physics, in
particular quantum field theory and string theory, to pure mathematics. In topology
this has led to new invariants of knots, three-manifolds and four-manifold. In
algebraic geometry this approach has transformed enumerative geometry. In this talk
I want to sketch the general philosophy behind this phenomenon, illustrate with
some simple examples the various mathematical ingredients, and point to some
recent developments that show that this line of research has not lost impetus. On the
contrary, modern ideas from string theory about dualities seem to point to vast
generalisations of what are by now well-established mathematical theories.
Jan 29 1999 Michael Atiyah (Edinburgh)
Euclidean 3-Space and the Unitary Group
A geometrical attempt to understand the spin-statistics theorem of quantum
mechanics has uncovered a surprising link between configurations of n distinct
points in Euclidean 3-space and the unitary group in n variables. This may generalise
to other Lie groups.
Jan 15 1999 Mike Lambrou (KCL)
Archimedes' $2,000,000 Manuscript
This was an informal talk on the story of a number of ancient mathematical
manuscripts that have survived to the present day. Some of them are extremely
valuable and have occasionally changed hands under dubious circumstances!
Dec 18 1998 Ton Coolen - KCL
`I learn, Mr. Fawlty, I learn ...!'
Conventional programmable computers are electronic civil servants. They can and
will do nothing that has not been explained to them in full detail. In the real world,
where data are messy and inconsistent, and where the number of possible situations
to be acted upon is infinite, there is no way of specifying all actions to be taken
beforehand, they are thus useless for decision making.
In contrast we nowadays find ourselves with the emergence of information
processing machines that `learn'. These start out being completely stupid, but
improve their performance gradually (i.e. rewrite their `program') by learning from
an expert `teacher'. In this talk I will give an informal and biased introduction to the
interesting mathematics emerging in the quantitative analysis of learning processes in
simple adaptive machines.
Nov 20 1998 Michael Berry - Bristol
Quantum Physics and the Riemann Zeros
The Riemann Hypothesis can be interpreted as saying that the prime numbers
contain `music' whose component frequencies are the Riemann zeros. The question
``Frequencies of what?'' leads to tantalising connections with the energy levels of
quantum systems whose corresponding classical motion is chaotic. At the level of
statistics, predictions for the Riemann zeros based on semi-classical quantum
asymptotics (with primes as periods of classical trajectories) have reached a high
degree of accuracy and refinement. For the zeros themselves, the Riemann-Siegel
formula and its improvements lead to new ways of calculating quantum levels. Much
evidence suggests that the operator that generates the zeros is related to the
harmonic oscillator in a sophisticated - but unknown - way.
Nov 6 1998 Jeremy Gray - Open University
Modernism in Mathematics
Around 1900, several of the leading mathematicians in different countries gave
popular explanations of their subject which are substantial in their discussions of
mathematics and address deep points of philosophy and psychology. They did so at
a time when geometry was becoming more abstract, and mathematicians were being
newly constituted as a profession. Profound fractures in the old formation of
mathematics appeared that challenged the practitioners and the broader public alike.
One measure of that challenge is the popular debate which flourished at the start of
the 20th Century about the nature of geometry.
Oct 23 1998 Tom Korner - Cambridge
Does Order Matter?
We know that the sum 1-1/2+1/3-..... can be made to converge to anything by
summing in the right way. But is `sensible summing' in real life subject to such
problems? We give examples from Fourier analysis and wavelet theory which relate
very directly to practical problems of signal storage to show that the answer is `yes
and no'.
Oct 9 1998 Norbert Schappacher - Strasbourg
``Who was Diophantus?''
The question ``Who was Diophantus?'' may today confront any mathematician if
for no other reason than that Fermat's so-called `Last Theorem' was originally a
marginal remark in his copy of Diophantus' Arithmetica. Mathematicians should
therefore have some basic knowledge of Diophantus. But in fact, we know next to
nothing about Diophantus the man - although this did not prevent historians of all
periods from speculating where solid information was lacking!
The main thesis of my talk will be that even the substance of Diophantus'
mathematics remains uncertain to a surprising degree. There are dramatic variations
in the way his Arithmetica have been read at different periods of mathematical
history, and although his work has been tremendously influential on more than one
occasion, it has never really been developed further. Diophantus' role was an
inspirational one, rather than that of a textbook. By looking at the various
``rebirths'' which Diophantus has undergone over time, I shall suggest a fresh
approach to this author today.