The Edinburgh Mathematical Society's President (Professor T A Gillespie) wrote in his Newsletter:
- The Lecturers of the St Andrews Mathematical Colloquium, 2003.
These were also part of the LMS/EPSRC Instructional Course 'Analysis and Probability on Fractals'.
The following gave lecture courses of five lectures:
(a) Professor Yuval Peres (University of California, Berkeley) - "Random Fractals."
(b) Professor Martin Barlow (University of British Columbia) - "Diffusions and Heat Equations on Fractals."
(c) Professor Wendelin Werner (Université Paris-Sud) - "Conformal restriction and related Questions."
There following three one-hour survey talks were given:
(d) Professor Kenneth Falconer (St Andrews) - "Probabilistic Methods in Fractal Geometry."
(e) Dr Lars Olsen (St Andrews) - "Fractals in Hyperbolic Geometry."
- Report by the Scientific Organiser Professor Kenneth Falconer.
This Short Course was held in the University of St Andrews from Monday 30th June to Saturday 5th July 2003. About 20 graduate students attended, with backgrounds from pure mathematics, probability, dynamics and theoretical physics. Sessions were held in the Mathematical Institute, and accommodation and meals were provided in David Russell Hall, some 15-20 minutes walk from the Institute. The Course was held in parallel with the 'St Andrews Colloquium' attended by about 15 self-supporting participants which allowed further academic and social interaction.
The Course presented contemporary areas at the interface of analysis, probability and fractal geometry, at a level aimed at research students with fairly broad ranging backgrounds. The material covered during the Course was very much as set out in the proposal.
The three main lecture courses complemented each other. For example, the theme of Brownian motion, which was presented in a classical form in the first lecture of Peres' course, provided a natural foundation for the rather recent ideas of diffusions on fractals and conformal restriction, which were discussed in the other two courses. Similarly, ideas such as Hausdorff and box dimensions and their calculation and properties cropped up in complementary forms in the three courses. The lecturers did an excellent job in putting across basic notions of the subject, but in a context which allowed an insight into recent and exciting research. Everyone agreed that the quality of lectures was extremely high, and we were fortunate to have three main lecturers who are at the forefront of world research in their areas, but who could also communicate so well.
Participants were provided with duplicated notes surveying basic ideas of fractal geometry (dimensions, iterated function systems, fractal measures, etc) prior to the course. Most of the lecturers circulated duplicated notes covering all or part of their talks. Some topics were illustrated by animated computer demonstrations.
The three main lecture courses each consisted of 5 lectures:
Professor Yuval Peres (University of California, Berkeley)
'Diffusions and Heat Equations on Fractals'
Professor Martin Barlow (University of British Columbia)
'Conformal restriction and related Questions'
Professor Wendelin Werner (Université Paris-Sud)
A synopsis of the material covered is appended.
Several tutorial sessions were held, with Dr Ben Hambly and Dr Toby O'Neil as tutors. These included review of problems set by lecturers, individual support and general discussion. With everyone resident in the same Hall and eating at the same table, the lecturers were accessible to the students, allowing a great deal of further informal interaction and discussion throughout the week.
There were three one hour survey talks, intended to give an overview of other topics on the theme of the meeting.
'Probabilistic Methods in Fractal Geometry'
Professor Kenneth Falconer (St Andrews)
Dr Lars Olsen (St Andrews)
'Fractals in Hyperbolic Geometry'
Dr Bernd Stratmann (St Andrews)
Participants were given the opportunity of contributing 15 minute presentations, though there was no pressure to do so. There were four such presentations, all on topics related to the main theme.
There were various social activities during the week to encourage a relaxed atmosphere. These included a guided tour of historic St Andrews, putting on the Himalayas, and an evening garden party.
Comments from participants and completed questionnaires suggested that the Course was very successful, not only from the instructional point of view, but also in integrating the research students into the national and international research environment.
- General Comments by Kenneth Falconer.
3.1. The number of UK research students attending was disappointing, despite considerable publicity efforts. One factor is that the number of research students in Britain in Analysis and Probability has fallen in recent years, partly because these areas are regarded as 'hard' at undergraduate level. It would probably have been possible to attract rather more students from the continent where analysis remains strong, with more effort in that direction.
3.2. The background of participants varied enormously, but on the whole the level assumed by the lecturers seemed a reasonable balance. We did our best to provide extra support and material for students with gaps in their background.
3.3. We were fortunate in having extremely friendly and approachable lecturers. The students very soon felt completely at ease with the lecturers and this encouraged informal interaction which contributed a great deal to the success of the meeting.
3.4. One organisational point: Several participants commented on the lack of a substantial gap between the first two morning lectures. However, the timetable was constrained to a considerable degree by the meal times available at the hall of residence combined with the 15-20 minute walk between the Mathematical Institute and the residence.
- Synopsis of Main Lecture Courses.
4.1. 'Random Fractals' - Yuval Peres.
Basic notations and definitions: Hausdorff dimension, capacity, packing dimension.4.2. 'Diffusions and Heat Equations on Fractals' - Martin Barlow.
Brownian motion, and its construction. Energy techniques for calculating dimension, applications to graphs of BM, image of sets under BM, and existence of multiple points. Fast points and Limsup random fractals.
Projections and energy methods for dimension, interiors of sets. Projections as a generalised concept, relations to BM, transversality. Sobolev dimension, conservation of energy under projection. Applications to distance sets, pinned distance sets and Bernoulli convolutions. Connections with harmonic analysis, algebraic numbers and dynamical systems. Open problems.
Graphs, random walks, percolation clusters - supercritical and critical cases. Connections between random walks and electrical resistance. Anomalous diffusion, exact fractals: the Sierpinski gasket.4.3. 'Conformal Restriction and Related Questions' - Wendelin Werner.
Random walks on the Sierpinski gasket. Volume growth (dimension), hitting times. Analytic inequalities - Poincaré and Harnack inequalities.
Transition densities on the Sierpinski gasket. Sketch proofs, properties of the continuum limit. Dirichlet forms and Laplacian. Heat equation on Sierpinski gasket and other fractals.
Random walks on percolation clusters - supercritical and critical cases.
Definitions of planar self-avoiding walks, critical percolation, survey on results and conjectures on their scaling limits
The conformal restriction property, example of conditioned planar Brownian motion.
Definition of SLE (Schramm-Loewner Evolutions), motivated by self-avoiding walks, first properties.
Conformal restriction property of SLE8/3. Relation between the outer boundary of planar Brownian motion, critical percolation clusters, and SLE8/3
Other equivalent constructions of the restriction measures. The Brownian loop-soup.
Intersection exponents between restriction measure samples. Relation with representation theory.