Final Report

EC FP5 Marie Curie Training Site: Banach Center for Differential Equations Functional Analysis and Stochastic Processes,  acr. BANACH, was hosted by the Institute of Mathematics of Polish Academy of Sciences and its international research and conference branch: Banach Center, in Warsaw, Poland.

The general objective was to provide a training for Young Researchers from Europe in several modern and challenging disciplines of mathematics and its applications: financial mathematics, modelling of biological processes, neural networks, stochastic control theory, stochastic partial differential equations, PDE's of fluids, dynamical systems, Banach and Sobolev spaces, complex analysis, theory of wavelets.  Fellows of the programme, PhD students in other (old) European Community states, were offered  a training through participation in seminars, courses, personal discussions with professors and local YR's, conferences in Banach Center. An objective was to provide them with a complementary knowledge and an additional stimulus to the own research and the future careers. They had access to one of the richest mathematical libraries in Europe as well as an opportunity to perform simulation and numerical modelling with the use of supercomputer facilities (due to a cooperation of Banach Center with the Interdisciplinary Center of Mathematical and Computational Modelling of Warsaw University).

BANACH hosted 14 fellows (17 stays), altogether 72 person-months, in 2001-2004, the length of visits between 3 and 12 months. some visits were extended, combined with other programmes, mainly EC Centre of Excellence: IMPAN-BC.There were at BANACH 5 fellows from Spain (Seville, Madrid), 3 from UK (Warwick, Belfast),3 from Germany (Bielefeld, Goettingen), 1 from Greece (Crete), 2 from France  (IRD Bondy, La Rochelle). The main seminars in which the fellows were involved were:Operator Theory, Functional Analysis, Dynamical Systems, Population Dynamics, Stochastic Processes and Stochastic PDE's.

16 publications, at least partially written at BANACH, published or submitted for publication were reported by the fellows, some of them in prestigious journals. Some of these papers are joint with Polish researchers. 8 fellows already defended PhD's (after stays at BANACH) and at least 5 of them, to our knowledge, stay in mathematics.

Some keywords referring to the results are:

spectrally bounded operators,  sections of a convex bodies, Volterra operators, integral operators, resolvents, square integrable martingales, stochastic integrals, stochastic PDE's, hedging in complete markets, Malliavin calculus, Banach function subspaces, spaces of analytic functions, local limit theorems, pseudo-Anosov homeomorphisms, Rouzy cycles, interval exchange maps, stochastic Fokker-Planck equation, travelling band solutions,   fish schooling, Markov semigroup, embedding in anisotropic Sobolev spaces, multifractal analysis.

 We had a very slow start, only two fellows during the first year, but later the programme became more popular and having more funds and time we could do much more. There were basically 2 types of fellows, finishing their PhD's and the beginners. It was a great pleasure that all! of them were truly involved in their studies.  We do believe that the participation in the programme has had and will have a substantial impact to their careers.  It contributed to the involvment of YRs in high level research. The fellows gained important personal international contacts with leading mathematicians and other YR's.

The programme contributed to the consolidating  the status of IMPAN -- Banach Center as an international research centre, to influence the scientific and technological progress of Poland and Europe.