Download Department of Mathematics Research Colloquia 2001 - 2002 Prof Tim Gowers Friday

Department of Mathematics Research Colloquia 2001 - 2002
Friday
Prof Tim Gowers
December Cambridge University
13 2002
Friday 8 Prof David Saad
November University of Aston
2002
Getting the message across - the statistical physics of error-correcting codes
Abstract:
Error-correcting codes are of significant practical importance as they provide
mechanisms for retrieving the original message after corruption during
transmission. We study low density parity-check error-correcting codes, using
methods adopted from statistical physics, to discover their typical theoretical
and practical limitations.
My talk will focus on the relevance of statistical physics to the study of errorcorrecting codes, different approaches that can be employed for carrying out
the analysis, theoretical and practical differences between various code
constructions and the insight gained from the analysis.
July 12 2002 Prof John Schwarz
Caltech
Supersting Unification
May 27
2002
Prof Misha Gromov
IHES Bures-sur-Yvette
Geometry of Configuration Spaces
Friday 15 Dr Adrian Kent
March 2002 University of Cambridge
Quantum linearity, relativity, and computability
Abstract:
Some time ago, Weinberg proposed testing the linearity of quantum theory
against nonlinear alternatives. This led Gisin and Polchinski to note the crucial
role linearity plays in allowing quantum theory to coexist peacefully with
relativity: nonlinear theories generally allow superluminal signalling through
entanglement. Conversely, the impossibility of superluminal signalling can be
used to derive quantitative bounds on the attainability of quantum information
processing tasks such as imperfect cloning. The computing power inherent in
nature, and the cryptographic uses of quantum information, also depend
crucially on the linearity or otherwise of the evolution laws: a non-linear
quantum computer would be far more powerful, while non-linear quantum
cryptography would be far more vulnerable.
In this talk, I review the relationships between linearity, relativity and
information processing, and point out the existence of a previously
unconsidered type of nonlinear theory compatible with relativity.
Friday 1 Professor John Cremona
March 2002 University of Nottingham
Rational Points on Elliptic Curves
Abstract:
The Arithmetic of Elliptic Curves used to be a subject only of interest to pure
mathematicians, even only to number theorists. Two rather different
applications of elliptic curves have brought them more into the public eye over
the last ten years: on the one hand, the connection with Fermat's Last
Theorem, which led to the famous proof by Wiles and others; and on the
other hand, their application in cryptography. (The latter application means
that most of us use elliptic curves, or will do soon, every day of our lives,
without knowing, whenever we use a credit card.)
My talk will cover more traditional ground. Given an equation with rational
coefficients, how can we determine whether or not it has any rational
solutions? Does it have finitely or infinitely many? How can we find them all?
Not one of these questions yet has a completely satisfactory solution! The first
half of the talk will assume no background knowledge, and give a survey of
what is known and what can be done in practice to solve specific equations.
Towards the end I will discuss some of the more recent techniques which have
been developed to help answer these questions.
Friday 8
February
2002
Professor Peter Whittle
University of Cambridge
"Recognition and Oscillatory Operation in Neural Networks"
Abstract:
This talk is concerned with both artificial and natural neural nets but no
knowledge is presupposed of either. An associative memory is a device which
`recognises' patterns in data, in that it assigns them to one of a number of
prescribed categories. We assert that such a memory must be able to cope
with `fading data' i.e. to form an inference even as its memory of the data
fades. A net deduced on this criterion shows striking biological parallels.
The other theme of this talk is the introduction of oscillatory operation, along
the lines suggested by W.J. Freeman. A number of remarkable effects emerge;
notably the modulation of the gamma-range oscillation by a slow square-wave
oscillation, reminiscent of the alpha-range oscillation evident in actual
electroencephalograms.
Friday 25 Professor John Cardy
January University of Oxford
2002
Counting Polygons and Lattice Trees in the Plane:
New Universal Results
Abstract:
I review recent work, partly rigorous and partly speculative, concerning the
counting of (1) self-avoiding polygons in the plane according to their
perimeter and area; (2) branched polymers or lattice trees according to their
mass. Theoretical physics notions such as supersymmetry and confinement,
and mathematical ideas such as $q$-algebraic functions, enter into the
arguments. The output is an exact scaling function which encapsulates
universal aspects of both problems. Its form is well-supported by lattice
enumeration studies.
Friday 14 Dr Giulia Iori
December King's College London
2001
"Statistical Mechanics Models of Consumption"
Direct interactions among economic agents, usually referred to as social
interactions (as opposed to market mediated interactions) are meant to capture
how the decision of each individual is influenced by the choice of others in his
reference group. In the economic literature the attention has been mainly
focused on the case of positive, pairwise symmetric, spillover, i.e. the case
where the payoff of a particular action increases when others behave similarly.
In a recent work, Cowan, et al. (1998), introduced a model of consumption
behaviour, hereafter called the CCS model, where the utility of an individual
agent is positively or negatively affected by the choices of other agents and
consumption is driven by peering, imitation and distinction asymmetric
effects. The CCS model has been analyzed in the framework of random utility
discrete choice models. These models have been analyzed using the
techniques of statistical mechanic. In the case of symmetric interactions, the
stationary or equilibrium probability can be expressed in the form of a Gibbs
distribution. With asymmetric interactions the long time behaviour of the
system has to be calculated by solving the dynamical problem and cannot be
evaluated by equilibrium ensemble averages. We use numerical simulations
with Glauber dynamics to explore the dynamical properties of the model.
Depending on the evolution algorithm, as well as the degree of the asymmetry,
the attractors can be either fixed points or limit cycles. We then introduce
noise in the system and study how this affects the dynamics of consumption.
Eventually, extending the analysis of CCS we discuss the role of costs and
memory in the consumption decision of the agents and consider different
scenarios for the connectivity among the economic agents.
Friday 30 Professor David Epstein
November University of Warwick
2001
"Defining and computing consensus trees"
By a weighted tree, we mean a finite tree where each edge in the tree comes
with a positive number, thought of as the length of the edge. For example the
tree might represent a portion of Darwin's "Tree of Evolution". Each vertex of
the tree, except the leaves, might represent a speciation event. The length of
an edge might represent the number of mutation events between the vertices.
Each leaf might represent an existing species. (Or each leaf might represent an
individual organism. Or each leaf might represent a bacterial strain---though
Darwin's tree is known to be a seriously wrong picture for bacteria.)
Billera, Holmes and Vogtmann (BHV) have produced a metric on the set of
such trees with specified leaves (www.math.cornell.edu/~vogtmann). In this
metric space, it is possible to give a rigorous definition of the average of a finite
set of weighted trees. (In fact one of the problems is that there are several
different concepts of average or centre in this space, each of which makes
mathematical sense.) BHV indicate an algorithmic approach to finding the
average. Their method is feasible if the number of leaves is small and the
number of trees is very small.
We have been working on constructing algorithms for analysing the BHV
situation, in such a way that the algorithms are feasible in the presence of
much larger sets of data.
Friday 16 Dr Martin Evans
November University of Edinburgh
2001
"45 Years of Directed Percolation"
Abstract:
The talk will give an overview of the problem of Directed Percolation first
introduced into the mathematical literature some 45 years ago: `Percolation
Processes' by S.R. Broadbent and J.M. Hammersley, Proc. Camb. Phil. Soc.
53 (1957) 629. The basic problem has reappeared in different contexts and in
different guises down the years, and examples of the many applications and
related systems will be presented. It will be stressed how the Directed
Percolation ``universality class'' is ubiquitous in the study of non-equilibrium
systems such as systems possessing an absorbing state. The talk will conclude
with an overview of open problems and questions of solvability of the system.
Friday 26 Professor Paul Townsend
October University of Cambridge
2001
"The Story of M"
Abstract:
For some time, efforts to arrive at a unified theory of all forces, incorporating
the Standard Model of elementary particle physics as well as Einstein's
gravity, have focussed on models of String Theory. Instead of point particles
in four dimensions, these theories use one-dimensional objects (strings) as
fundamental constituents, propagating in a ten-dimensional space-time.
Recent progress in this area has shown that string theories in fact contain
further objects, so-called branes, some of which are higher- dimensional and
which are as fundamental as the strings themselves. Using branes, unexpected
relations among different string models could be established,as well as relations
to supergravity in eleven spacetime dimensions. This suggests that there is a
yet more fundamental theory, dubbed M-theory, underlying ten-dimensional
string theories and eleven-dimensional supergravity. The talk aims at telling
"L'histoire d'M", as far as it is known today, to a rather general audience. The
challenge posed to M-theory by an accelerating universe is to be discussed
briefly at the end.
Friday 12 Professor Frances Kirwan
October (University of Oxford)
2001
"Group valued moment maps"
Abstract:
The concept of a moment map (or momentum map) in symplectic geometry is
a generalisation of the familiar notions of angular and linear momentum in
mechanics, and has been studied for several decades. It is a smooth map from a
symplectic manifold X to the dual of the Lie algebra of a group G acting on X,
whose components are Hamiltonian functions for the infinitesimal action on X
of elements of the Lie algebra. A few years ago Alekseev, Malkin and
Meinrenken introduced the concept of a quasi-Hamiltonian G-space, for
which there is a moment map taking values in the group G itself instead of in
the dual of its Lie algebra. The aim of this talk is to describe some of the
similarities and differences between group valued moment maps and
traditional moment maps, and an application of the new approach.
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 ChurchTuring 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 MTheory. 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.