Network title:
Network short title:
The principal contractor and the members listed below shall be jointly and severally liable in the execution of work defined in part B of this annex:
The principal contractor
Université Bordeaux 1, established in France
The members
Vrije Universiteit, Amsterdam, established in the Netherlands
Universitat Autonoma de Barcelona, established in Spain
University College Dublin, established in Ireland
University of Leeds, established in the United Kingdom
Université Pierre et Marie Curie, established in France
The Norwegian University of Science and Technology, established in Norway
Vienna University of Technology, established in Austria
Tel Aviv University, established in Israel
St Petersburg department of the V.A. Steklov Mathematical Institute, established in Russia
The principal contractor and the members are referred to jointly as ``the participants''.
1) Function Theory For Bergman and related spaces, develop factorization theory and characterize ``inner-outer'' functions in terms of their growth near the boundary. In one-dimensional and multidimensional situations, characterize interpolating and sampling sequences, and use interpolating Blaschke products to find new invertibility criteria for Toeplitz operators.
Develop approximation theory (harmonic approximation on general sets, tangential approximation), and find further relations between quadratic identities and best approximation problems. Improve understanding of capacities in metric and geometric terms, and, more generally, improve understanding of the Cauchy and Hilbert transforms.
2) Operator Theory Develop the theory of function models, and clarify the relation between spectral properties of Hankel and Toeplitz operators and function theoretical properties of their symbol. Develop new operator theoretical methods to analyse problems arising from concrete classes of integral differential delay equations. Describe the spaces spanned by generalized eigenvectors for nonselfadjoint operators arising from delay equations. Find new applications of the Brown approximation scheme, and use function theoretical tools to study bi-invariant subspaces of l2ω(Z).
3) Geometry of Banach spaces, Convex Geometry Develop the aspects of the geometry of Banach spaces already directly related to function theory and operator theory, and continue the ``transfer of technology of Banach spaces'' to these areas. Find new applications to analysis, convex geometry and statistical mechanics of the principle of concentration of measure and the majorizing measure theorem. Improve estimates for contractive approximation algorithms of convex bodies by polytopes. Develop variational principles and pursue their applications to differential equations.
All research in mathematics is based both on the development of powerful abstract methods and on the detailed study of important problems. A specific point in the network is the importance of the interplay between methods, results and problems arising from real and complex analysis, operator theory and the geometry of Banach spaces. This interplay is the basis of the whole research problem and its methodology. For example, the recent counterexample to Halmos' 10th problem involves Hankel operators, the theory of martingales and results from the geometry of Banach spaces. The theory of contractive divisors G on the Bergman space starts with Hilbert space methods, but uses potential theory to produce a smooth function Φ vanishing on the unit circle such that the Laplacian of Φ agrees with |G|2-1 in the unit disc, and developments of this theory are based on the fact that the biharmonic Green function for the disc is positive. Recent progress on the description of the translation invariant subspaces of the weighted spaces l2ω(Z) is based partly on the theory of almost analytic functions and Dynkin estimates and partly on the theory of entire functions of exponential type, and the Brown approximation scheme is also a potential powerful tool to make further progress in this area. Also the interplay between operator theory and complex function theory is the basis of the network's methodology to analyse problems arising from various classes of integral, differential, difference and delay equations.
The following division of tasks is based on already existing collaborations between the teams and on reasonable progress expectations by the participants. For all these tasks the teams directly involved in the research are indicated between brackets.
TASK 1: HARDY AND BERGMAN SPACES, INTERPOLATION (1, 3, 4, 7, 9, 10)
COORDINATOR: PARTICIPANT 7
It is expected at the end of the four years to understand ``inner'' and
``outer'' functions in the Bergman space in terms of growth at the boundary,
and to be able to exhibit noncyclic vectors without zeroes in all
weighted Hardy spaces. It is expected to reduce the gap between
necessary and sufficient conditions for zero-sets in the Bergman space,
and to characterize interpolating and sampling sequences for a large
class of spaces defined in terms of derivatives.
TASK 2: CAUCY INTEGRAL, CAPACITIES, HARMONIC APPROXIMATION (1, 3, 4, 7, 10)
COORDINATOR: PARTICIPANT 4
It is expected at the end of the four years to approach complete understanding
of analytic capacities in terms of Ahlfors-David regular sets and Minkowski
dimension. It is also expected to obtain significant progress on boundary
behaviour of functions in the Bloch class and for conformal mappings,
and new progress on harmonic, tangential approximation and thin sets.
TASK 3: FUNCTION MODELS AND APPLICATIONS OF OPERATOR THEORY
(1, 2, 5, 7, 8, 9, 10)
COORDINATOR: PARTICIPANT 2
The workplan consists in developing specific methods (the band method
for Nehari-Takagi problems, the state space method for inverse spectral problems, etc.)
and also to make ``effective'' powerful abstract methods.
For example the theory of function moels provides powerful tools,
the characteristic function, the minimal dilation, but to investigate
the spectral structure of a specific contraction one has to compute
them explicitly. An ``effective'' Gelfand theory, in which when possible
estimates are given for inverses, can be developed in a similar way for
specific algebras. During the four years an impetus on applications of
operator theory from progress in these directions is expected.
TASK 4: THE INVARIANT SUBSPACE PROBLEM
(1, 6, 7, 9)
COORDINATOR: PARTICIPANT 9
The Bordeaux team will be in charge of extending the Brown approximation
method to commuting pairs of contractions, and in the case of
absolutely continuous contractions with isometric functional
calculus of finding what is the most general extension of Bourgain's
factorization theorem for H1. The four teams
will try to fill the gap between the results obtained by
nonquasianalytic and quasianalytic methods to obtain a very general
existence theorem for nontrivial translation bi-invariant subspaces
on l2ω(Z) for nonincreasing weights, and
in some situations a complete description of them.
TASK 5: GEOMETRY OF BANACH SPACES AND APPLICATIONS
(1, 5, 6, 7, 9, 10)
COORDINATOR: PARTICIPANT 10
Team 6, in relation with operator theorists in the network, will
study important similarity questions arising from the theory of
operator algebras. Teams 5 and 6 will also pursue investigations on
the boundedness of the vector-valued Cauchy transform and on the
related question of closedness of the sum of two closed, resolvent
commuting operators. Teams 6 and 10 will try to extend various real
interpolation results, like K-closedness of the couple
(H∞(T),Hp(T)) in
(L∞(T),Lp(T)) to some
multidimensional situations and Hp-like spaces,
in relation to the Grothendieck property. Teams 1 and 6 will keep
developing nonsmooth analysis for smooth Banach spaces
and try to determine for general Banach spaces a notion of solution
for Hamilton-Jacobi equations extending the notion of viscosity solutions
for smooth spaces.
TASK 6: CONVEX GEOMETRY, CONCENTRATION OF MEASURES (6, 9)
COORDINATOR: PARTICIPANT 6
Teams 6 and 9 will pursue investigations in the flatness problem
for lattice-point-free convex bodies, using Young's inequalities
and their converse, and try to develop a theory of K-convexity
for nonsymmetric convex bodies. They will also apply the theory of
concentration of measure to sudy random matrices and to try to formalize the
Sherrington-Kirkpatrick model in statistical mechanics and some models
of memorization.
The expected progress at the end of the four years is indicated above for each task. For each task the following milestones indicate goals by which the progress can be assessed, either at the time of the Mid-Term Review (24 months) or at the time of the final report (48 months).
Milestones
TASK 1
1) Existence of noncyclic vectors for all wighted Hardy spaces in the
log-convex case (48 months).
2) Characterization of complete interpolating sequences in weighted Hilbert spaces
of entire functions (24 months)
3) Characterization of discrete random fields for which entire functions
with zeroes in the field belong almost surely to the Fock space (24 months).
TASK 2
1) Applications of capacities to approximation problems in spaces
of analytic and harmonic functions (48 months)
2) Characterization of exceptional sets at the boundary for
harmonic functions (24 months)
3) Computation of best constant inequalities for conjugate harmonic functions
(24 months).
TASK 3
1) Explicit computation of the characteristic function for new classes
of contractions, and obtain resolvent tests for similarity to a normal
operator (24 months).
2) Use of the band method to describe the set of all solutions for some
Takagi-type interpolation problems (48 months)
3) Use of the Agler-Young operator-theoretical method to make progress
on the μ-synthesis problem in engineering (24 months).
TASK 4
1) Reflexivity for pairs of commuting contractions with dominant Harte
spectrum (24 months).
2) Description of the lattice of invariant subspaces of the Bergman
space counterpart of the Fourier binest algebra (48 months).
3) Complete description of translation bi-invariant subspaces of
l2ω(Z) for a large class of
nonincreasing weights not bounded away from zero (24 months).
4) Reflexivity properties of the shift operator on
l2ω(Z)
in the case of thick spectrum (48 months).
TASK 5
1) Examples where Pisier's similarity degree satisfies 3 < d <∞
(48 months).
2) Comparison between regularity and λ-regularity for sums of
operators in UMD-spaces (24 months).
3) Formula for the Fréchet subdifferential of the product of two positive
lower semicontinuous functions in smooth Banach spaces (24 months).
TASK 6
1) Improvement of the Kannan-Lovasz estimate O(n2)
for the flatness of n-dimensional bodies (24 months).
2) Explicitation of a direct link between the Sherrington-Kirkpatrick model
and questions related to the group of isometries of Rn,
n very large (48 months).
The remainder of the annex deals with administrative details that are not of general interest.
Last updated November 23rd 2002 by Jonathan Partington