Plenary Lecture
Prof. Robin C. Ball
Saturday
12:30-13:30, MS02
Department of Physics and Centre for Complexity Science, University of Warwick
Optimisation under Uncertainty
It is a fact of life: we have to choose between complex options whose outcome is uncertain.
I will focus on the following scenario:
- the space of choices is too large to search exhaustively;
- for each choice we do know the probability distribution of the outcome;
- we have agreed what figure of merit we are trying to optimise, but we can only etimate
this (with statistical errors!) by sampling from the distribution of outcomes. Three examples
are how to design a protein molecule to spontaneously fold as fast as possible; how to plot a
route of least expected length in advance of some stops becoming redundant (Probabilistic
Travelling Salesman Problem); how most profitably to position the wells and pipelines to an
oil field, before you know detail of where the oil will flow underground, the future price of
oil, and installation costs. The problem is that the more accurately we evaluate each choice,
the fewer choices we can explore. I will show how Simlulated Annealing by (Markov Chain)
Monte Carlo can be adapted to deal with these problems. Simulated Annealing is directly
analogous to cooling a physical system slowly, high merit corresponding to low energy in the
final state. The key idea is that using rough estimates of the merit mimics an effective
temperature: initially it is better to use rough estimates than more precise ones. The
resulting technique has been termed Stochastic Annealing, and for known probability
distribution of the errors the correspondence to true thermal equilibrium sampling can be
made precise. This work was developed with (then!) research students Thomas Fink (now at
Institut Curie, Paris) and Neil Bowler (now at UK MetOffice) [1]. Finally I will discuss a more
open-ended example where we desire to optimise the worst likely case. The classic
example is in Financial Portfolios, where this is expressed as the Value at Risk: in practice an
arbitrary level of risk is adopted, say 1%, and we seek to minimise the Value there is 1%
chance of losing. Stochastic Annealing can be used directly on this problem. However I will
also discuss how an adaptation of it does something more intriguing: it optimises the value
which appears to be at 0% risk on the basis of a limited depth of sampling of the distribution
of outcomes. Thus instead of setting an arbitrary standard of risk, we set a limiting depth of
search. This came out of work with Masters student Jonathan Mascie-Taylor [2].
[1] R.C. Ball, T.M.A. Fink, and N.E. Bowler, Stochastic Annealing, Physical Review Letters, 91,030201 (2003).
[2] J. Mascie-Taylor, MSc Project, Centre for Complexity Science, University of Warwick (2009).
Keynote Lectures
Combinatorics
Prof. Peter Cameron
Thursday
14:00-15:00, MS03
School of Mathematical Sciences, Queen Mary, University of London
Synchronization
The topic of synchronization arose in automata theory, but has connections with
permutation groups and graph homomorphisms. I will discuss some of these, and also an
application of Joyal's proof of Cayley's Theorem on trees to the synchronization behaviour of
a random single-transition automaton.
Mathematical Biology
Prof. Raymond Goldstein
Thursday
14:00-15:00, MS04
DAMTP, University of Cambridge
Fluid Dynamics and the Evolution of Biological Complexity
One of the most fundamental issues in evolutionary biology is the nature of transitions from
unicellular organisms to multicellular ones. Many basic questions arise in this context: What
is the advantage of being larger? What are the driving forces behind the appearance of
distinct cell types? In this talk I will describe an approach to these broad questions based on
the use of a particular lineage of green algae which serves not only as a model for
evolutionary studies, but also for biological fluid dynamics. Experimental and theoretical
results will be described that focus on the scaling laws for nutrient uptake as a function of
size, particularly as affected by fluid flows driven by the flagella of these organisms, the
synchronization of those flagella, and the mechanism by which multicellular organisms
composed of thousands of cells exhibit accurate phototaxis in the absence of a central
nervous system.
Mathematical Physics
Prof. Sandra Chapman, FInstP
Thursday
14:00-15:00, MS02
Physics Department, University of Warwick
Scaling laws, emergence and statistical descriptions of systems that are out of equilibrium:
what we can model and measure
Physics aims to find descriptions of natural systems that are universal. From the starting
point of dimensional analysis, through simple CA models that show emergence, to what can
be observed in data, we will explore out of equilibrium systems with examples including
astrophysical turbulence, laboratory plasmas, ecology and neuroscience. We will pose the
question- what are the prospects for universal theories of out of equilibrium systems that
can be tested on real world data?
Differential Geometry
Prof. Nigel Hitchin, FRS
Thursday
14:00-15:00, MS03
Mathematical Institute, University of Oxford
Generalized geometry and Poisson geometry
Generalized geometry is a differential geometric structure which seems to contain naturally
some of the features of supersymmetric theories but can be viewed in very concrete terms
and in particular one can draw on analogies with ordinary complex or Riemannian geometry.
On the other hand the role of Poisson structures, especially holomorphic ones, also turns
out to be an important aspect. The talk will discuss the interrelationship between the two,
and how ideas from one can suggest approaches in the other.
Complexity
Prof. Jeff Johnson
Thursday
17:30-18:30, MS03
Open University
Hypernetworks for modelling multilevel complex systems
Hypergraphs generalise graphs by allowing edges to be sets with any number of of vertices.
Although a big step forward, hypergraphs are set-theoretic constructs with anomalies such
as {D, O, G} = {G, O, D}. Hypernetworks overcome this by the definition of 'relational
simplices', <a, b, c, ...; R> where the n vertices a, b, c, ... are related under the n-ary relation
R. For example, <mum, dad, son, daughter; R_family> forms a social construct called a family.
Here the object with four vertices is represented as a 3-dimensional tetrahedron, a higher
dimensional analogue to the 1-dimensional edges in networks. This example also illustrates
how n-ary relations can define different levels of aggregation, with the parts being at lower
levels to wholes. Relational simplices have a higher dimensional connectivity structure, for
example two tetrahedra sharing a triangular face are 2-connected, a higher dimensional
analogue to edges being 0-dimensionally connected by vertices in networks. Simplices carry
patterns of numbers over their vertices and faces that can represent dynamics such as flows
or system activity. These dynamics are different to the structural dynamics of the formation
and disintegration of relational simplices. In this talk I will show how hypernetworks of
relational simplices can represent relational structure and dynamics in complex multilevel
systems. The emphasis will be on ideas rather than technical details, and illustrations will be
given from social, physical and engineering systems.
Algebra
Prof. Martin R. Bridson
Friday
15:00-16:00, MS05
Mathematical Institute, University of Oxford
The universe of finitely presented groups
Symmetry is everywhere in mathematics and the language in which to describe and examine
symmetry is that of group theory. How should one describe (infinite, discrete) groups? How
should one go about understanding them? Can one ultimately hope to understand all
(finitely presented) groups completely, or will undecidability phenomena necessarily intrude
at some point? Can one use geometry to understand an arbitrary group, and if so, what
geometry? Can one hope to understand infinite groups by understanding all of their finite
images? Lots of questions! In this talk, I'll describe the broad outlines of the modern study of
infinite discrete groups, sketch the universe of finitely presented groups, present answers to
some of the preceding questions, and point to some of the most exciting directions of
current research.
Financial Mathematics
Prof. David Hand, FBA
Friday
15:00-16:00, MS04
Department of Mathematics, Imperial College
Evaluating consumer credit scoring models
Predictive statistical models are universally used in the consumer credit industry to guide
decisions about who should be offered financial products such as loans, credit cards,
mortgages, car finance, and so on, and also to monitor the behaviour of customers to see if
they are running into difficulties and to detect fraud. In this talk I examine such models, with
a particular emphasis on their evaluation: accurate evaluation is critical to ensure that
correct commercial decisions are made. Some recent results are presented.
Statistics and Probability
Prof. Saul Jacka
Friday
15:00-16:00, MS02
Department of Statistics, University of Warwick
Stochastic Control and Applications
Stochastic Control is a slightly neglected weapon in the probabilist's (and occasionally the
statistician's) armoury. The talk will focus on some basics of stochastic control and some
representative applications.
Topology
Prof. Caroline Series
Friday
15:00-16:00, MS03
Warwick Mathematics Institute, University of Warwick
Limits of limit sets
A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Its limit set is the set
of points where orbits accumulate on the boundary of hyperbolic space, which we identify
with the Riemann sphere. As the group varies continuously, we explore how limit sets can
degenerate, even collapsing to a space filling curve. Is it true that limit sets always move
continuously with the group?
Algebraic Geometry
Prof. Burt Totaro, FRS
Saturday
10:15-11:15, MS02
DPMMS, University of Cambridge
Algebraic geometry from a topological point of view
We discuss the role of topology in algebraic geometry. There are simple but surprising
arguments which show that many properties of a complex submanifold of projective space
are controlled by its topology, in fact by the second homology group. More subtle properties
are determined by a convex cone, the "cone of curves". For algebraic surfaces, the cone of
curves can be studied using hyperbolic geometry.
Number Theory
Prof. Kevin Buzzard
Saturday
11:30-12:30, MS02
Department of Mathematics, Imperial College
Future directions in the Langlands program.
The Langlands program is a profound series of conjectures about analytic gadgets called
automorphic forms. Some of these conjectures relate these analytic gadgets to much more
algebraic gadgets (elliptic curves, Galois groups and so on). More recently there has been
some serious attempt to try and formulate p-adic versions of these conjectures, which
should in some sense be easier because p-adic analysis is far more "algebraic" than real
analysis. I will attempt to give some sort of an overview of both the classical, and the more
modern p-adic, theories. Be warned: I have to start somewhere, and I don't want to spend
30 minutes on basic definitions and properties of the p-adic numbers, so I will spend under 5
minutes on such things and then press on. On the other hand I will give precise definitions of
things like automorphic forms and not expect the audience to be familiar with Langlands
program at all.
Analysis and PDEs
Sir John Ball, FRS
Saturday
11:30-12:30, MS03
Mathematical Institute, University of Oxford
Variational problems for solid and liquid crystals
The talk will describe how the calculus of variations can provide insights into the behaviour
of solid and liquid crystals, focussing on some open problems.
Dynamical Systems
Prof. Mary Rees, FRS
Saturday
11:30-12:30, MS04
Department of Mathematical Sciences, Liverpool University
Topological models in complex dynamics
A general problem in complex dynamics is to describe a dynamical system up to change of
coordinates, that is, up to (at least topological) conjugacy. This is an especially big topic in
complex dynamics, when a large variety of examples is easy to exhibit in a single parameter
space. I shall describe some of the common constructions, and some of the problems that
arise.