Wspólnotowy Serwis Informacyjny Badan i Rozwoju - CORDIS

Final Activity Report Summary - HANAP (Harmonic analysis, nonlinear analysis and probability)

The main objective of the project was to transfer knowledge from western Europe to the members of the Institute of Mathematics of Wroclaw University in order to bring new ideas, modernise, enlarge and unify the scope of research in harmonic analysis, probability theory and nonlinear partial differential equations.

The unifying theme of the project was harmonic analysis related to randomness and nonlinearity. Around 60 mathematicians of all levels, including faculty members, graduate students and the best research oriented undergraduate students, with half of the students being female, participated in the programme that comprised of:

1. harmonic analysis methods in hydrodynamics,
2. harmonic analysis related to group actions,
3. non-commutative probability,
4. stochastic and harmonic analysis methods in queueing networks, and
5. higher order difference operators.

Twenty one foreign researchers were employed by the HANAP project and each of them stayed at the university for at least two months and gave 14 to 15 lectures of 90 minutes duration each. They spoke about various subjects, such as harmonic analysis tools for evolution equations, Navier-Stokes equations, pseudodifferential models in continuum mechanics, random walks, ergodic theory, in particular ergodic theory of non commuting random products, spectral distributions of discrete Laplace operators, spherical harmonic analysis on affine buildings, special functions, analysis and probability related to root systems, discretisation of non-commutative random variables, non-commutative martingale inequalities, random matrices, asymptotic theory for representations and characters of symmetric groups, spectra of growing graphs, moment problems in orthogonal polynomials, point process modelling, stochastic geometry, discrete time stochastic networks and analysis of rare events in systems with heavy tails.

Each course counted for six credit points and both graduate and undergraduate students could choose them as a part of their studies' programme. The courses were split into an introductory part, accessible to a broader audience, and a more specialised part. Following a general introduction into the analysed subject, the latest progress in the field was presented. Lecture notes were also available. Problem solving sessions were arranged as part of the lectures.

The programme resulted in 283 publications. The main scientific results were:

1. a thorough analysis of the chemotaxis model of the Keller-Segel featuring diffusion with a nonlocal drift and its various extensions
2. asymptotic behaviour (heavy tail) of the stationary measure of matrix and, more generally, Lipschitz recursions
3. proofs of related limit theorems with stable laws being limits
4. asymptotics of the invariant measure of an affine recursion in the critical case
5. asymptotics of the characters of irreducible representations of the symmetric groups that corresponded to balanced Young diagrams and their application to the analysis of computational complexity of some quantum algorithms
6. asymptotics of representations of a given fixed Lie group in the limit as the typical highest weight tended to infinity
7. study of fluctuations of random matrices in the limit as their size tended to infinity
8. investigation of fluid networks driven by multidimensional Levy processes
9. asymptotics of the ruin probability in various insurance models
9. study of time and space correlations for queuing networks.

The project lasted for were 46.5 incoming and 28 outgoing person months altogether. Twelve researchers from Wroclaw participated in extensive visits at partner institutions. We finally organised 13 conferences. Trough extended visits of foreign mathematicians, intense training and scientific exchange HANAP heavily stimulated the integration of Wroclaw team within the European Research Area and considerably increased its research potential. The members of the group not only benefited from the 'west Europe' knowledge but also trained graduate students coming from there.

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