RESEARCH
SEMESTER



Geometric Methods
in Analysis and Probability

Organisers:

J. B. Cooper (Linz)
P. W. Jones (Yale)
V. Milman (Tel Aviv University)
P. F. X. Müller (Linz)
A. Pajor (Université de Marne-La-Vallée)
D. Preiss (University College, London)
C. Schütt (Kiel)
C. P. Stegall (Linz)

Erwin Schrödinger Institute, Wien
May-August 2005

Final report: pdf, Postscript

List of participants

Recommended accomodation


FOR FURTHER INFORMATION PLEASE CONTACT

Institute for Analysis, Section Functional Analysis
renata@bayou.uni-linz.ac.at


[Print version]

We are organizing a research semester on the above theme, for the period May - August 2005. It will consist of the following three sections:
  1. Geometric Aspects of Infinite Dimensional Banach Space Theory
    Nonlinear Methods in Linear Functional Analysis
    The main activities will be concentrated in the period May 25th - June 5th.
    (Organisers James Cooper, Linz, David Preiss, London, and Charles Stegall, Linz)
  2. Conformal Invariance, Probability and Singular Integrals
    The main activities will be concentrated in the period June 10th - July 5th.
    (Organisers Peter W. Jones, Yale and Paul F. X. Müller, Linz)

  3. Asymptotic Theory of the Geometry of Finite Dimensional Spaces
    The main activities will be concentrated in the period July 10th - August 5th.
    (Organisers Vitali Milman, Tel-Aviv and Carsten Schütt, Kiel)


1. Geometric Aspects of Infinite Dimensional Banach Space Theory
Nonlinear Methods in Linear Functional Analysis

The main topics will be

Differentiability properties of Lipschitz and convex functions, non-linear theory (isomorphism problems), geometric measure theory and Hausdorff dimension, porosity, generalizations of Baire category theory, abstract Haar measure and other ideas of small sets. Recently even techniques from logic have been applied in this area; for example there were four different classes of convex functions whose differentiability properties were under investigation and it has been shown (mostly due to Kalenda) that it is not possible to prove that any of these are equal.

Although enormous progress in these areas has been made in recent years, as documented by the authoritative monograph of Yoav Benyamini and Joram Lindenstrauss, there remain many problems; in particular the following are of importance:

  1. Given two Lipschitz functions on $\ell^{2}$, do they have a common point of Frechet-differentiability?
  2. Let $Y$ be a space with Radon-Nikodym property. Does there exist a $\sigma$-porous set $E$ in $\ell^{2}$ and a Lipschitz map $f:\ell^{2}\rightarrow Y$ such that all the points of Gateaux-differentiability of are contained in $E$?
  3. Let $M$ be a separable metric space and $\mu$ and $\nu$ Borel measures on $M$. Suppose that for all balls $B$ and $C$ in $M$ we have $\mu(B\cap C)=\nu(B\cap C)$. Are $\mu$ and $\nu$ necessarily equal? (The answer is no if one has only equality on balls.)
  4. Let $G$ be an abelian topological group with a translation invariant metric. Let $\mu$ and $\nu$ Borel measures on $G$ such that for all balls $B$ we have $\mu(B)=\nu(B)$. Are $\mu$ and $\nu$ necessarily equal?
Another goal would be to examine in detail the algorithmic aspects of the work of D. Preiss on differentiability. Ideas using porosity and abstract Haar measures have been used for this purpose. We expect that the discussions between the experts in these diverse areas will be very productive.

2. Conformal Invariance, Probability and Singular Integrals

The meeting on Conformal Invariance, Probability, and Singular Integrals will focus on recent and spectacular solutions to long standing open conjectures in these fields.

We list four theorems around which the meeting will center:

  1. The Hausdorff dimension of the planar Brownian frontier is 4/3. Lawler, Schramm (Salem Prize Recipient), and Wendelin Werner (European Mathematical Society Prize) proved this conjecture of Mandelbrot and established the importance of a new probabilistic/analytic method:The Stochastic Loewner Equation (introduced earlier by Oded Schramm). This result led to a new field studying the geometry of SLE traces.

  2. The proof of Cardy's formula and conformal invariance of critical percolation clusters by S. Smirnov (Salem Prize and Clay Prize). This result turns out to be very strongly tied to SLE, though the methods developed are quite different. Along with topic 1, this work has had a deep impact on Conformal Field Theory.

  3. The proof that analytic capacity is semiadditive. X. Tolsa (Salem Prize) solved the corresponding conjecture from the 1950s, of Vitushkin, by solving the (more general) Melnikov conjecture.This work establishes sharp geometric criteria for the boundedness of the Cauchy Singular Integral Operator.

  4. The proof of Calderon's Conjecture on the boundedness of the bilinear Hilbert Transform. Due to Lacey and Thiele (Salem Prize Recipients), this breakthrough opened the door to recent theorems on Multilinear Singular integrals, established by Muscalu, Thiele and T. Tao (Salem Prize and Clay Prize). The underlying methods established by these authors provide a deep understanding of the phase plane geometry as pioneered by analysts such as A.P Calderon, L. Carleson, R.R. Coifman and Y. Meyer. Calderon had made this conjecture to explain corresponding behavior of the Cauchy Integral Operator.

The main topics of the program will be the items of this list and their interactions. While apparently covering disjoint subjects, the fact is that scientific interaction between these topics is the main point of the meeting. We describe briefely how the various areas developed historically: Building on work in Hardy Spaces, Brownian Motion, and Dynamical Systems from the past forty years, a community grew up sharing a common background in techniques as well as philosophy. In general the philosophy was to study some sort of geometry (whether of phase-plane analysis, singular integrals on curves, harmonic measure, dynamically defined sets, etc.) by using the correct multi-scale analysis. During the past twenty years these areas have seen explosive growth, with many new techniques being invented. While the scope of investigations has both changed and expanded over the years, the various communities have maintained strong ties to each other. The following topics will form the main focus of the program: Percolation; Self-avoiding Random Walks; Stochastic Loewner Equations; Cardy's Formula and applications to Conformal Field Theory; Brownian motion; the Cauchy Integral Operator; Analytic Capacity; Calderon-Zygmund operators on non homogenous spaces; Multilinear Singular Integrals; uniform estimates for Paraproducts.

During 2001/02 the Mittag-Leffler Institute devoted a year-long program to the topic CONFORMAL MAPPINGS and PROBABILITY. The proposed meeting at the Schrödinger Institute is designed to be both a follow up and partial extension of that program. The 1999 meeting at the Schrödinger Institute, organized by the Functional Analysis group of the J. Kepler University should also be considered as a precursor to the current activities.

Colloquium in Stift Schlägl, June 24-26, 2005:   List of Participants, Talks

3. Asymptotic Theory of the Geometry of Finite Dimensional Spaces

The main topics will be concentration of measure phenomena, transportation of measure, asymptotic theory of convex bodies, geometry of high-dimensional normed spaces. In 2002 a semester devoted to this topic was organised at the Pacific Institute for Mathematical Sciences in Vancouver. One of the organizers was V. Milman. People from different areas were brought together including computer scientists from Microsoft. This semester proved to be very successful. A conference at the Schrödinger Institute 3 years later would allow us to examine the status of the progress and initiate new research. It is very important to invite young reseachers to the meeting.

In the last 20 years, probabilistic methods have proved to be extremely efficient in studying geometric properties of convex bodies in high dimensional spaces. They have been used e.g. in proving results on the regularization and symmetrization of convex bodies, volume estimates for convex bodies, in Dvoretzky-type theorems, in sampling theory or in the approximation of general convex bodies by polytopes in some metric or in a volume sense, using random and deterministic methods.

Typical effects in high dimensions are often connected with concentration of measure phenomena which by now have been used not only in the theory of Banach spaces and convex bodies but also in other rather diverse areas of mathematics such as complexity theory, classical probability theory or discrete mathematics.

Probabilistic methods have yielded geometric inequalities for convex bodies which translate into the estimation of various parameters such as type/cotype, general volume ratios or Banach-Mazur distances, in particular to Euclidean spaces. We propose to extend this approach to open problems on the approximation by polytopes, to the study of random polytopes, to describing interesting specific norms defined by convex bodies, to the realization of concrete Euclidean subspaces of certain Banach spaces and to extensions of Khintchine-type inequalities which are relevant tools in the subject and are related to this study of the random structure of the volume distribution of convex bodies.

We mention some specific results on convex bodies which have been proved by probabilistic and geometric methods in the last years in order to demonstrate the effectiveness of these methods. Results of [BLM] state that any symmetric convex body $K$ can be transformed uniformly isomorphically into an Euclidean ball by $t$ rotations, $\frac{1}{t}\sum_{i=1}^{t}T_{i}(K)$ where $t\sim(d(K)/\omega(K))^{2}$, $d(K)=$ diameter of $K$, $\omega(K)=$ mean width of $K$. The rotations $T_{i}$ are generated probabilistically, but the result is also best possible, up to a factor, in a deterministic sense. This indicates a deep and unexpected connection between achieving almost the best deterministic result and by achieving it by probabilistic methods. Far reaching extensions of this result may be found in [LMS].

Another aspect is approximation problems concerning convex bodies. How well can a convex body be approximated by a polytope with a restricted number of vertices, faces, flags or similar quantities? There are many things known by now, although a lot of questions are still open. These questions are related to problems in tomography, image processing, data transmission and problems in computer science. Some invariants used in image processing are from affine differential geometry [SaT].

Talks and preprints


References

[BLM] J. BOURGAIN, J. LINDENSTRAUSS, V. D. MILMAN, Minkowski sums and symmetrizations, Geometric aspects of functional analysis (1985/86), 82-95, Lecture Notes in Mathematics 1267, Springer, Berlin, 1987.
[LMS] A. E. LITVAK, V. D. MILMAN, G. SCHECHTMAN, Averages of norms and quasi-norms, Math. Ann. 312 (1998), 95-124.
[MS] V. D. MILMAN, SCHECHTMAN, Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Math. J. 90(1997), 73-93
[SaT] Sapiro G., Tannenbaum A. (1994): On invariant curve evolution and image analysis. Indiana University Journal of Mathematics, 42, 985-1009



Past ESI Semesters:

This page is part of a site using frames. Start your visit from the main page.
Comments are welcome by email
Renata Mühlbachler