EuroSummer School

Instructional Conference on
Combinatorial Aspects of Mathematical Analysis

Edinburgh, 25 March to 5 April 2002

Scientific Programme | Conference Arrangements | Application Forms | Conference Home Page

Scientific Programme

The conference will start on the evening of Monday 25th March with an informal buffet combined with Registration at Pollock Halls.

Speakers details and course titles are given below, together with any abstracts and suggestions for further reading. Clicking on an speaker's name within the timetable will take you directly to the relevant part of this page.

0930-1030 1100-1200 1330-1430
(Tutorials to 1500)
1500-1600 1630-1730
Tue 26 Carbery I Kalai I Ledoux I Barthe I Kalai II
Wed 27 Carbery II Ledoux II TUTORIAL Kalai III Odell I
Thu 28 Ball I Bollobas I TUTORIAL Tao I Odell II
Fri 29 Tao II Bollobas II TUTORIAL Kalai IV Odell III
Sat 30 Tao III Bollobas III      
Mon 1 Christ I Green I TUTORIAL Giannopoulos I Tao IV
Tue 2 Christ II Green II TUTORIAL Schechtman I Christ III
Wed 3 Giannopoulos II Green III TUTORIAL Schechtman II Beckner I
Thu 4 Barthe II Ball II Giannopoulos III Schechtman III Beckner II

Lunch will be provided at 12.15 on each full day (Tuesday to Friday in week 1, Monday to Thursday in week 2). Refreshments will be available every morning between 1030 and 1100, and all afternoons (except Saturday) from 1600 to 1630.

Keith Ball (UC London)
Brunn-Minkowski and reverse isoperimetric inequalities.
The Brunn-Minkowski inequality is one of the cornerstones of the study of the geometry of convex bodies and also plays a crucial role in information theory. The first talk will explain the idea of the inequality and show how it implies the classical isoperimetric inequality in Euclidean space. The second talk will address the reverse isoperimetric problem for convex bodies using a powerful convolution inequality from harmonic analysis which will be proved in the talk of Franck Barthe.

Suggested Reading:
Chs 5 & 6 (Ball) Flavors of geometry, ed. Silvio Levy, CUP
Chs 1 & 3 also useful preparatory reading.

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Franck Barthe (Marne la Valle)
Transportation and functional inequalities
The aim of these two lectures is to present the Brascamp-Lieb inequalities, a powerful extension of the well-known Hölder's inequalities, together with a proof using the solution of an optimal transportation problem. The Brascamp-Lieb inequalities have applications in harmonic analysis (they are linked with famous inequalities by W. Beckner) and in geometry (K. Ball will present applications to the geometry of convex sets). On the other hand the transportation problem "how to carry some supply from where it is to where it is needed, in a most efficient manner, given a tranportation cost depending on the displacement?" is an old subject, addressed by Monge, which was recently revived and found applications in various areas of mathematics.

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William Beckner (U Texas, Austin)
Geometric Inequalities on Manifolds
The focus of these two lectures will be to outline how one can both adapt classical Euclidean analysis and develop new analytic arguments to provide insight for geometric structure on Riemannian manifolds. Sharp constants for functional inequalities over a manifold encode geometric information. Asymptotic arguments identify geometric invariants that characterize large-scale structure. This approach is illustrated by discussion of the topics:
  1. isoperimetric inequality
  2. analysis on hyperbolic space
  3. symmetrization
  4. convolution estimates on non-unimodular groups
  5. asymptotics and convexity relations for embedding constants.
Beckner, Geometric inequalities in Fourier Analysis [1995]
Beckner, Sharp inequalities and geometric manifolds [1997]
Beckner, Geometric asymptotics and the logarithmic Sobolev inequality [1999]
Beckner, On the Grushin operator and hyperbolic symmetry [2001]
Chavel, Riemannian Geometry [see chapter 6 on isoperimetric inequality] [1993]
Gromov, Isoperimetric inequalities in Riemannian manifolds [1986]
Gromov, Metric structures for Riemannian and non-Riemannian spaces [see chapter 6 and appendix on Paul Levy's isoperimetric inequality] [1999]
Hebey, Nonlinear analysis: Sobolev spaces and inequalities [1999]

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Bela Bollobás (Cambridge and Memphis)
Phase Transitions in Large Combinatorial Systems
Phase transitions are of great interest in diverse fields, including statistical physics, probability theory, discrete mathematics, and computer science. In spite of much effort, there are very few rigorous results; in most cases we have only conjectures based on computer experiments. In these three lectures we shall present precise results about phase transitions in a variety of basic random discrete structures. The fundamental results concern the sudden change in the component structure of a random graph as the number of edges passes through a certain threshold. These results, going back to the work of Erdös and Rényi forty years ago, were the very first rigorous results about phase transitions. We shall also present recent results obtained with Borgs, Chayes, Kim and Wilson about the exact nature of the phase transition in the random 2-satisfiability problem, and describe results of Friedgut and Bourgain about the speed of phase transitions in general systems.

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Anthony Carbery (Edinburgh)
Occurrences of combinatorial problems in harmonic analysis
Classical Harmonic Analysis is turning out to be a rich source of interesting problems with a flavour of geometric combinatorics. In these lectures we discuss a few of these problems, how they arise in harmonic analysis, and some results to date. Many open problems remain. The emphasis will be on the interaction of the areas of harmonic analysis and geometric combinatorics and the aim is that everything will be elementary and from "first principles", although some willingness to take things "on faith" for those not trained in harmonic analysis will be desirable.

Possible topics:
  1. Besicovitch, Kakeya and Nikodym sets and their associated maximal functions; their roles in harmonic analysis; some of the combinatorial issues arising. This would be an introduction to the lectures of Tao.
  2. Oscillatory Integrals and Sublevel Set problems: some combinatorial issues.
  3. Combinatorial problems suggested by the restriction phenomneon.
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Michael Christ (U of California, Berkeley)
Elementary counting techniques and Lp operator estimates
One preoccupation of harmonic analysis during recent decades has been the detailed study of estimates, in Lp spaces, for various linear operators whose properties are heavily influenced by the presence of some form of curvature. A second theme, which has attracted increasing attention in the last few years, is the Lp analysis of certain multilinear extensions of standard linear operators.

In these lectures I will discuss two particular problems, illustrating each of these two overarching themes. A unifying feature is that elementary combinatorial techniques, which is to say, arguments that amount merely to counting points, have thus far yielded insights which seem not to be accessible via Fourier transform-based techniques. Both these problems concern operators which are defined by integration against nonnegative kernels, yet are not completely trivial to analyze.

Through a discussion of these specific examples I hope to illustrate larger themes and interconnections.

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Apostolos Giannopoulos (U of Crete, Heraklion)
Geometry of the Banach-Mazur compactum.
Click here to download the postscript file of this abstract

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Ben Green (Cambridge)
Structure theory of set addition
We will discuss a theorem of Freiman, which states that a subset of Z which is roughly closed under addition is contained in a "box" of small dimension. This theorem is fundamental to the work of Gowers on Szemerédi's theorem. We hope to give a recent proof of this theorem due to Chang, which modifies an earlier method of Ruzsa. In so doing we will introduce a range of concepts from combinatorial number theory, particularly
  • Plunnecke's inequalities
  • Minkowski's second theorem and Bohr neighbourhoods
  • Freiman homomorphisms
  • Introductory Fourier analysis on ZN
  • The role of dissociativity in Fourier analysis
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Gil Kalai (Hebrew University of Jerusalem)
Harmonic analysis of Boolean functions
We will discuss results, problems and applications concerning Fourier analysis of Boolean functions. Among the topics:
  • Relations with discrete isoperimetric inequalities
  • The notions of influence of variables and noise sensitivity
  • Threshold phenomena for Boolean functions
  • Relations with complexity theory
  • Applications to probability, percolation theory and social choice theory.
The course will be elementary and self-contained

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Michel Ledoux (Toulouse)
Concentration, transportation and functional inequalities
We present and discuss aspects of the concentration of measure phenomenon from the geometric, measure theoric and functional points of view. In particular, emphasis will be put of functional semigroup tools to reach and investigate concentration properties. Some recent applications will also be discussed.

Suggested reading:
The concentration of measure phenomenon Mathematical Surveys and Monographs, AMS 2001

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Ted Odell (Texas A&M)
Ramsey methods in Banach spaces
  1. Introduction to Ramsey theorems, Banach spaces and the interplay between the two.
    We will be focusing on the use of Ramsey theory to the study of the isomorphic structure of separable infinite dimensional Banach spaces X. Questions that occur are often of the form: given a space X (perhaps of a particular type) and a property (P) does there exist a subspace Y having (P)? In order to solve them one must sometimes stabilize a certain function defined on a structure to some substructure and this is where Ramsey theory enters the picture. We'll present some Ramsey theorems and give some necessary Banach space background.
  2. Spreading Models and other applications
    We will define a spreading model E of a Banach space X and show how its existence is guaranteed by Ramsey theory. E is a Banach space which yields asymptotic information about X and sometimes about the subspace structure of X. We will present some known results and open questions and a generalization of spreading models.
  3. Gowers' Dichotomy Theorem
    We will present W. T. Gowers' famous "block Ramsey" theorem and its main application to Gowers' famous Dichotomy Theorem.
Our goal is to make the lectures accessible to people with a solid background in basic functional analysis.

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Gideon Schechtman (Weizmann Institute)
Tight embeddings and concentration inequalities
Click here to download the .dvi file of the abstract.
Suggested reading:
Schechtman, Concentration , results and applications (Mostly sections 3.1 and 3.2) Click here to download the postscript file
Johnson and Schectman, Finite dimensional supspaces of Lp (mostly section 2.1), in Handbook of the Geometry of Banach spaces, Vol 1 W. B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam (2001), 837-870 Postscript file available - Click here

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Terence Tao (UCLA)
Recent progress on the Kakeya family of conjectures
Define a Besicovitch set to be a subset of Rn which contains a line segment in every direction. The Kakeya conjecture asserts that these sets always have Hausdorff dimension n. This conjecture is still open in three and higher dimensions, and is related to several outstanding problems in oscillatory integrals, Fourier summation, number theory and PDE. We will survey these connections and describe some recent results.

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