The Colloquia are relaxed and non-technical talks aimed at being accessible to all the postgraduate students.

A list of current colloquia is available here

Apr 27 2001   Prof. Roger Penrose     (Oxford)

`Gravitational Quantum State Reduction'

Mar 23 2001   Prof. Henri Gillet     (Illinois)

`Connections between solving Diophantine equations and analysis on manifolds

 

Abstract:

I will give a survey of some of the work that has been done (since the early 80's) on this topic, with emphasis on the analogies that inspired much of the work. Starting with the conjectures of Vojta (inspired by the Mordell conjecture, and analogies between Diophantine questions and Nevanlinna theory), through the work of the speaker and Soulé on arithmetic intersection theory (extending the ideas of Arakelov), and the work of Zhang. I hope to include some discussion of open problems.

Mar 16 2001   Dr. Andrew Hodges     (Oxford)

`Uncomputability in the work of Alan Turing and Roger Penrose'

 

Abstract:

This talk will be mainly historical, charting in some detail what Alan Turing meant by 'machine' between 1935 and his last work in 1954, and how the meaning of 'machine' is related to his famous assertions about the prospect of machine intelligence. Penrose's critique of 'Turing's Thesis' is an important stimulus and reference point for this review.

Mar 2 2001   Prof. Brian Davies     (King's)

`Infinite Machines'

 

Abstract:

We describe in some detail how to build an infinite computing machine within a continuous Newtonian universe. The relevance of our construction to the Church-Turing thesis and the Platonist-Intuitionist debate about the nature of mathematics are also discussed.

Feb 16 2001   Prof. Caroline Series    

(Warwick)

 

`Slicing spaces of Kleinian groups'

 

Abstract:

A Kleinian group is a discrete group of linear fractional transformations which acts on the Riemann sphere. This action extends to hyperbolic 3 space which can be modelled as the interior of the ball. Discreteness implies that there is a region (a fundamental domain) which is moved around disjointly from itself by the group. Usually we can find a suitable region on the sphere, but in transitional cases it may only exist in hyperbolic 3 space.

As we vary the complex valued parameters defining the generators of the group, there will be transitions between discrete and non-discrete groups. Typically, this boundary has extremely complicated fractal behaviour, and pinning it down is intimately bound up with deep questions about hyperbolic 3-manifolds.

We shall describe how the first computer pictures of such boundaries were made and some recent developments, both theoretical and practical. (Illustrated with lots of pictures!)

Jan 19 2001   Prof. Garry Gibbons    

(DAMTP, Cambridge)

 

`Convex Cones in Physics

 

Abstract:

The idea of a convex cone is a very simple one but nevertheless it has a surprisingly large number of applications in physics and deserves to be better known. In this lecture, which is intended to be introductory, with no previous knowledge of cones assumed, I hope to show how the language of convex cones arises naturally in General Relativity, Quantum Mechanics and Supersymmetry. In General Relativity it relates to causal theory, positive mass theorems,the dominant energy condition, ergo-regions, Killing Horizons and black holes. The state spaces of quantum mechanics are convex cones and they also appear in the classification of BPS states in supersymmetric theories. Homogeneous cones and Jordan Algebras are closely related and their symmetries and conformal symmetries may play a role in M-Theory. To see how these symmetries act and what they mean, especially the conformal symmetries, it is useful to generalise the ideas of Special and General Relativity to those of Finsler Geometry a generalisation which is interesting in its own right. Some natural examples involve a novel generalisation of the dynamics of matrices which have cropped up in studies of twistor particle mechanics. Finally the complexification of convex cones leads to a consideration of generalised future tubes. One possible application is to holography and the AdS/CFT correspondence.

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 undreamt-of 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.