STARTPAGE

HUMAN RESOURCES AND MOBILITY (HRM)
ACTIVITY

MARIE CURIE ACTIONS

Marie Curie Host Fellowships for the Transfer of Knowledge (ToK)
Development Host Scheme

PART B

 SPADE2
♠♠
DETERMINISTIC AND STOCHASTIC DYNAMICS, FRACTALS, TURBULENCE

B1. Scientific Quality of the Project

B1.1. Research topic (general description)
The research topics include: mathematical modelling of various processes (physical, biological, societal, economical), by ordinary and partial differential equations (ODE's, PDE's), deterministic and stochastic,  long time  behaviour of trajectories, scaling and aggregation limits, studies of local geometric (fractal) structure of arising attractors. The challenges concern the complexity of phenomena and their models as well as the structural richness of mathematical methods used. The research content of SPADE has horizontal and vertical structure. It is a coherent package of areas and methods in mathematics and mathematical physics, of great importance in applications.
Areas (horizontal) include
a) Finite (low dimensional) dynamical systems (flows and iteration), in particular: 1D-dynamics, real and complex Iterated Function Systems (IFS) as a kernel of higher dimension phenomena (hyperbolic, non-uniformly hyperbolic, dominated splitting). A spectacular progress in 1D iteration has been achieved by D.Sullivan, J.-Ch.Yoccoz (Fields medal in 1994), C.McMullen (Fields medal in 1998) in renormalization and J.Graczyk, G.Swiatek, M. Lyubich and very recently O.Kozlovski, W. Shen, S. van Strien in density of hyperbolicity. This development is leading to better understanding of dimensions and measures on metric attractors and basins.
b) Infinite-dimensional systems. We will focus on turbulence in 2D (Navier-Stokes Eq.), attractors, Sinai-Ruelle-Bowen (SRB) measures and multifractal spectra (J.Bricmont, A.Kupiainen, S. Kuksin), turbulence in 3D thin domains, with engineering applications in lubrication (P.Constantin, C.Foias, R.Temam).
c) Scaling limits in physical processes. Since eighties theoretical physicists predicted exact values of many dimensions and scaling exponents for the 2D models like percolation, Ising model, self avoiding random walk, etc. Recently works on Schramm-Loewner evolution led to rigorous mathematical confirming of these predictions. We expect advances in the models like diffusion limited aggregation (a generic model of fractal growth) or random matrices (of major importance in studying disordered media).
Methods include
1. Ergodic Theory and Information: time averages and statistics along trajectories, equilibrium measures, mixing, etc.
2. Thermodynamical Formalism and Multifractal Spectra of Lyapunov exponents, entropy, dimensions. The subject was started in the sixties and seventies (SRB), and it was further
developped in the eighties by Procaccia, Kadanoff, et al. It remains in a close relation with geometric measure theory. We want to learn the applications at INRIA Rocquencourt “projet fractale”.
3. Evolution Stochastic Processes (also as areas of research). Evolution Stochastic ODE’s and PDE’s. 
4. Methods of Function Spaces: Sobolev spaces, functions of bounded variation, analysis of singularities of measures, singular integrals and harmonic analysis. Wavelet methods.
 Some of these fields are strong in Poland, in particular at IMPAN, but have become too specialised, and suffer somewhat from isolation and brain drain. Other fields, of primary importance, are at the beginner stage, but we do want to develop them. We have started to integrate researchers working in the fields by organising joint seminars. Acquiring fresh ideas and methods from leading research groups in the world should give us new strength. It will also mobilise Polish researchers to conduct interdisciplinary investigations and to apply the mathematical tools in other fields. The ToK programme would help to maintain IMPAN as an international research centre.
 B1.2. Project objectives

We describe the scientific objectives according to the tasks (see B1.4). Partner organisations and/or names of some external researchers involved will be listed in B1.4 and further on.
Our overall objective is to increase the transfer of ideas and modern mathematical tools to Poland and (through the Banach Center) to Central and East Europe, by intensifying the contacts with the strongest centres in Europe and outside. We want to concentrate on the following scientific objectives.
1. Dynamical Systems. A progress in description of classes of rigid and stable chaotic (weakly hyperbolic) and quasi-periodic dynamics and deformations, modelling various processes. Exploration of function theory (quasi-conformal), topological-combinatorial methods, geometric measure theory (thermodynamical and multifractal formalisms) tools. Development of applications in population dynamics (blood cells, etc), small scale (fractal) structures. Some areas of applications (e.g. multiscale phenomena analysis, biomedical problems) are already investigated at the ICM of Warsaw University (IMPAN partner).
1a. Applications of fractals and IFS.  We want to learn at INRIA “projet fractale”.  An objective is to become involved  in their “fractals and engineering” programmes, 1D-signals (computer network traffic analysis, speech synthesis), 2D-signals (image analysis, denoising and segmentation, image compression).   
2. Partial Differential Equations, turbulence and asymptotics
Many features of the physical phenomenon of turbulence in fluid flows can be explained rigorously in the frame of the theory of infinite dimensional dynamical systems. We are interested in finite dimensionality of the flow of two-dimensional Navier-Stokes (NS) fluid (e.g. on two dimensional torus) and three dimensional NS fluid in thin 3-D domains (Constantin, Foias, Temam, Robinson, Sell, Raugel, etc.). The number of degrees of freedom of such flows is connected to the Hausdorff dimension of the global attractor. Dimension estimates given in terms of the Grashof number correspond to the Kraichnan and Kolmogorov dissipation lengths that concern classical descriptions of turbulence. We are also interested in such close problems as long time behaviour of solutions of semi-linear evolution equations, regularity of solutions in Gevrey classes, the study of models of pattern formation, blood motions and plane motions.
2a. Turbulent transport. Transport of pollutants, heat, chemical agents or magnetic fields plays an important role in environmental issues, meteorology, engineering or astrophysics. Rough models of passive turbulent transport based on random dynamical systems permit to understand the basic phenomena like occurrence of cascades of conserved quantities, and presence of intermittency characterized by anomalous scaling laws and multifractality. Such models provide a bridge between hydrodynamics and the dynamical systems theory stretching the latter to the situations (fully developed turbulence) where the usual tools break down. 
Exploration  IFS approximations for transport phenomena of practical importance: high Reynolds number flow and porous medium flow in multiscale materials.
3. Stochastic Processes. The main aim of stochastic PDEs, is the study of qualitative properties of solutions. In particular long time behaviour, existence of invariant measures and attractors, ergodicity, large deviation estimates, exit probabilities and large noise asymptotics. The stochastic turbulence applied to systems of interacting particles, stochastic Navier-Stokes, Euler and Burgers equations and Anderson models in two or three dimensional regions is
of special interest. These equations appear naturally in fluid mechanics as a description of the dynamics of a fluid under an influence of a random external force. We plan also to learn and start working in Growth Models (exhibiting fractal phenomena) and Stochastic Loewner equation, SLEκ. The need for the study of Stochastic Processes appears in the investigation of chaotic ODE (deterministic chaos).
4. Methods of Function Spaces.  Isomorphic properties of function spaces and their dependence on smoothness and domain; singular integral operators acting on the Sobolev spaces; the use of interpolation in the study of PDEs; multidimensional wavelet bases, approximations by splines in the spaces of smooth functions, Sobolev spaces on metric spaces (e.g. Carnot-Caratheodory spaces).

B1.3. Impact to the host organisation

In our project we choose several complementary topics and objectives. As the general impact, we expect that:
(1) The new techniques and results developed at foreign Universities and Institutes will be transferred to Poland (to researchers at IMPAN and at the Universities - through common seminars, research groups and advanced courses).
(2) Different research groups will concentrate their effort on common or closely related problems. For example attractors in PDE's will be investigated by methods of function spaces, geometric measure theory, ODE's approximations, information theory etc.. 

International state-of-the-art and contribution to the advance of capabilities of the applicant, by research fields.

1. Dynamical systems. Many investigations have been encompassed by the general Thom-Smale-Palis programme, to study typical dynamics and prove that in many situations most of the space consists typically of basins of attraction to periodic motions, to describe separating chaotic sets and study changes of the patterns with changing parameters.  
In 1D real iteration the Density of Hyperbolicity has been proved recently in Warwick, opening new perspectives due to new methods. In complex case (rational, entire or meromorphic) the problem is still open, attracting attention of leading specialists. Related is the problem of Local Connectedness of the Mandelbrot set (MLC), demanding better understanding of the renormalizations. Warsaw had one of top research groups in 1D iteration, but suffered heavily from the emigration of strongest people (Misiurewicz, Nowicki, Swiatek, Graczyk, Urbanski). Some of them maintain close contacts with us (joint papers) and ToK would encourage some to return. IMPAN Katowice group working in population dynamics with the use of 1D and stochastic processes methods will also get a new impetus . 
In higher dimensions the theory of limit sets of IFS is emerging, especially in non-conformal case (solenoids). (Self) intersecting Cantor sets are of this nature. They appear in the parameter space in the homoclinic tangency bifurcation (Palis, Yoccoz, Moreira). In applications of IFS we want to benefit from the cooperation with INRIA "projet fractale" (see B1.2, 1a). 
Recently the theory of dominated splitting, generalising hyperbolic (or Anosov) systems,
is developing (Pujals, Bonatti). We plan to join the mainstream due to a cooperation with  the Brasilian, French and Portuguese schools.  There has been major progress in understanding equilibria, in particular SRB (Sinai-Ruelle-Bowen) measures and their multifractal spectra on
invariant sets  (attractors).         

2. PDE's, turbulence, asymptotics. It is well known (the Leray millennium problem) that the three dimensional Navier-Stokes equations (NSE) do not constitute a proper dynamical system. Thus to obtain a proper dynamical system one is confined to two dimensional problems or three dimensional problems in thin domains. Such problems are natural in some applications, (e.g., in lubrication theory). Moreover, in realistic models the domains and the boundary conditions considered do not usually fit the existing theory, which concentrates on simplified problems (e.g., considering homogeneous boundary conditions and sufficiently smooth domains).
 In the project we focus on the above problems. Our research concerns studies of flows in thin domains and relation between time and space asymptotic of the flow (between dynamics of NS-flows in thin 3D domains and 2D dynamics of Reynolds flows considered in the theory of lubrication). This new field of research involves applications of dynamical systems to lubrication theory.  We also include the NS motions in pipes with large velocity which could model the motion of blood in vessels. In Europe there is a very active group working on uniqueness and regularity for NSE (Raugel, Cannone, Meyers, Penel, Wiegner, Sohr, Neustupa, Pokorny). Using the programme ToK we shall intensify cooperation with these researchers, learning from them and working together. A closer collaboration with Braaksma, Balser and Gramchev will allow us to develop further the method of re-summation of formal solutions and apply it to nonlinear evolution equations such as semi-linear heat, KdV and shallow water equations.
3. Stochastic Processes.  Intensively studied are distributed control systems both deterministic and stochastic. We believe that the analytic methods developed in our Institute (Zabczyk, Peszat) for uncontrolled models would be applicable. We would like to collaborate on control of flexible and reaction-diffusion systems with Pisa, Detroit and French Flight Centre (A. Bensoussan). Stochastic differential geometry, in particular stochastic equations on manifolds, construction of a Gaussian measure on infinitely dimensional manifolds (loop spaces), are important, yet a little neglected subjects in Poland. They have fascinating applications and, in particular, they are directly related to sub-Riemannian geometry. Collaboration with University of Warwick (D. K. Elworthy), University of Nancy (R. Leandre) will help to develop this subject in Poland. Interacting particles systems, percolation, processes, super-processes and interplay between discrete and continuous models are of great importance. They can be used as models of various physical phenomena and they are important tool for studying stochastic PDE's. We expect to profit from the collaboration with R. Tribe (Warwick) and J. F. Le Gall (Ecole Normal Sup. Paris). We plan to learn Dirichlet forms technique to construct Markov processes in relation for example with potential theory, in collaboration with University of Bielefeld (M. Röckner), University of Bonn (S. Albeverio).
3a. Scaling limits in physical processes. Recently there was significant progress in the mathematical understanding of random conformally invariant objects in the plane. Lawler, Schramm, and Werner proved Mandelbrot's conjecture on the dimension of the Brownian frontier being 4/3, established values of Brownian intersection exponents predicted by the physicist Duplantier, and constructed the scaling limit for the Loop Erased Random Walk. Kenyon has proved a number of conjectures for dimer models, made by physicists. The predictions of the physicist Cardy have been proved for the critical percolation on triangular lattice by Smirnov, who also constructed its continuum scaling limit. Johansson obtained remarkable results on relations between two-dimensional random growth models and random matrix theory. Carleson and Makarov provided a deterministic model, via Loewner Equation, of diffusion limited aggregation (DLA) (Witten-Sanders model) getting Kesten (lattice) model type estimate of growing cluster.
4. Methods of Function Spaces. A remarkable progress was achieved in the last few years around the fast developing subject of wavelet expansions. We are going to develop wavelet methods in numerical solutions of differential equations and non-linear approximation – collaboration with A. Shadrin (Cambridge). The connections between the measure theory, non-linear functional analysis and the theory of differentiation of functions, including functions on infinitely dimensional spaces, received a lot of attention in recent years - we are going to collaborate with D. Preiss (University College London) in this domain. Theory of spaces of smooth fuctions (non-reflexive Sobolev spaces, spaces of functions of bounded variation, anisotropic spaces)  require deeper study and further development. This includes Fourier and wavelet analysis, embedding and trace theorems (V. Kolyada), Sobolev spaces on metric measure spaces.

B1.4. Research method and work plan

Tasks, scientists, organisations. The ToK programme is divided into four linked tasks. We list (in brackets) the names of local researchers involved in the tasks. In square brackets we list some foreign scientists potentially involved in the project. They will be consulted during the recruitment process (some were consulted already).
1.  Deterministic Finite Dimensional Dynamical Systems. (F. Przytycki, M. Rams, R. Rudnicki, M. Wojtkowski, A. Zdunik, J. Kotus);
•	Real 1-D dynamics and complexification, analytic and combinatorial theory, geometric measures, multifractal point of view [A. Douady, J. Graczyk - Orsay; J.-Ch. Yoccoz, College de France, S. van Strien, O. Kozlovski - U. Warwick, G. Levin – Jerusalem, G. Swiatek- Penn State, K. Astala, P. Koskela, P, Mattila –U. Helsinki, X. Buff – Bordeaux, K.L. Petersen, B. Branner – Copenhagen, M. Shishikura –Kyoto, J. Rivera-Letelier -Antofagasta] ;
•	Hyperbolic and dominated splitting, non-uniform hyperbolicity: Lorentz and Henon attractors, Sinai-Ruelle-Bowen measures, homoclinic tangency bifurcations, limit sets for non-conformal iterated function systems, selfintersecting Cantor sets [M.Benedicks - KTH Stockholm, J. Palis, M. Viana, H. Pujals - IMPA Rio de Janeiro; S. Luzzatto – IC London, V. Baladi - Paris 6, J. Schmeling - U Lund; K. Simon - Tech. Uni. Budapest ]. (Possibly also topics in hamiltonian and geodesic systems, chaos between quasi-periodic motions. [S. Kuksin - Edinburgh (also Task 2); P. Le Calvez - Paris 13; H. Eliasson: Paris 6; A. Chenciner -Paris 6, J. Llibre, C. Simo - Barcelona, R. de La Llave - Texas; Marmi - SNS Pisa.]).
2. Partial Differential Equations, turbulence, asymptotics.  (W. Zajączkowski, G. Łukaszewicz, J. Rencławowicz, A. Kałamajska, T. Regińska, G. Łysik ,).
•	Infinite dimensional dynamical systems and lubrication theory. Atractors, SRB-measures,  mixing [G. Raugel, R. Teman – Paris 6, J. C. Robinson – U Warwick, G. Bayada – INSA, Lyon, A. Kupiainen –U. Helsinki, J. Bricmont –Lourain, V. Baladi – Paris 6];
•	 Equations of viscous fluids: existence and regularity for Navier-Stokes [G. Raugel, M. Cannone, Y. Meyer - Paris, P. Penel - Toulon, M. Wiegner - RWTH Aachen, H. Beirao da Veiga - Pisa, A. Sequieira - Lisbon, G. Prause – Milan, W. Schroeder - RWTH Aachen]; 
•	Evolution equations, asymptotics, [T. Gramchev - Cagliari U., W. Balser – U. Ulm]; 
3. Stochastic Processes. (A. Lasota, T. Komorowski, S. Peszat, R. Rudnicki, L. Stettner, J. Zabczyk)
•	Scaling limits and conformal invariance of physical processes, percolation and superprocesses, diffusion limited aggregation Schramm-Loewner evolution, random matrices [W. Werner - Paris-Sud, R. Tribe - U. Warwick, C. Muller - Rochester, K. Burdzy - Seattle, K. Gawedzki – ENS Lyon, Carleson, Benedics, K. Johansson –KTH Stockholm, , Smirnov KTH & Geneve];
•	Turbulent transport [K. Gawedzki –ENS Lyon, D. Bernard – CEA Saclay, M. Oliver – IU Bremen];
•	Stochastic geometry: stochastic ODE on manifolds and the geometry of path spaces [D.K. Elworthy - U. Warwick], stochastic equations on infinite dimensional manifolds [R. Leandre - Nancy, Z. Brzezniak - Hull];
•	Stochastic control [J-L. Menaldi - Detroit, A. Bensoussan - French Flight Center, M. Tessitore -Milano, L. Tubaro - Trento, G. Da Prato - SNS Pisa)]; 
4. Methods of Function Spaces. (A. Kamont, P. Mankiewicz, P. Wojtaszczyk, A. Pełczyński, M. Wojciechowski);
•	Isomorphic properties of function spaces, interpolation, singular integrals, Carnot-Caratheodory spaces [G. Pisier – Paris 6, Garcia-Cuerva – U. Autonoma Madrid, V. Kolyada – U. Karlstat, Brudnyj, Shwartzman - Technion Jerusalem, P. Jones - Yale, P. Mueller - Linz , Vogt -WUP, M. Roginskaya – Goteborg, D. Preiss, M. Csörnyei – UCL, G. Alberti  -Pisa, A. Volberg -Paris6];
•	Wavelets and approximation, unconditional structures, non-linear methods [A. Shadrin - Cambridge, DeVore, N. Kalton - Columbia Mo., Terenzi - Milano];

Some motives present in all the tasks. 
1. The project is designed to focus some attention on population dynamics which is applied for example in genetics, oncology, epidemiology and ecology. A.Lasota and R.Rudnicki collaborate with the Centre of Oncology – Gliwice in mathematical modelling of tumor growth, their spacial structure and the optimization of chemo- and radiotherapy, they will participate in FP6 RTN on this topic – recently invited for negotiations,  see also B6.  
 In the theory of population dynamics we use the methods of stochastic processes, differential equations and functional analysis. For example in studying ecological models we use two approaches: the Eulerian viewpoint in which we are only interested in the distribution of some characteristic of the population and the Lagrangian one where the behaviour of each member of the population is taken into account. Models in the first approach are partial differential equations and we are interested in such properties like existence of equilibrium states, stability, bifurcations, travelling waves and chaos. To study these properties we use spectral analysis, semigroup theory and dynamical systems. The Lagrangian approach is based on stochastic processes (diffusion and point processes) and computer simulations. The passage from Lagrangian approach to Eulerian one leads to fascinating mathematical questions connected with stochastic partial differential equations. 
2. To investigate Stochastic Evolution it is necessary to develop techniques based on nonlinear functional analysis, in particular on the theory of nonlinear dissipative operators and function spaces, like Sobolev, Besov and interpolation spaces, on partial differential equations including viscosity solutions and on harmonic analysis and stationary random fields. Rapid development has taken place in the theory of Kolmogoroff operators originating from stochastic PDE's and in the study of their associated semigroups.
3. Remarkable is the interweaving of deterministic and random methods/phenomena. Randomization helps to solve problems hard in deterministic models. Often stochastic models are more relevant. Probabilistic models describe deterministic (chaotic) dynamics. On the other hand there appear deterministic models parallel to random (e.g. diffusion limited aggregation,
see  B1.3, 3a)
4. In the root of dynamical systems there are methods of ergodic theory (Birkhoff Ergodic Theorem), Information Theory (Shannon) Mixing properties, thermodynamical formalism 
and multifractal spectra linking measures to geometry.

Schedule Each of the 4 tasks will have 2 semesters of concentrated work, including seminars, courses and lectures. However since the tasks are closely interrelated, the activities of each task will continue during the whole life of the programme (and beyond). They will be combined with (usually preceded by) basic courses by local staff for students at Warsaw University and Warsaw University of Technology and PhD students from IMPAN. These will prepare the students for participation in more advanced activities delivered by the ToK guests. Planning 66 person/months visits of experienced and more experienced researchers during the programme, we expect to have 2 visitors from abroad at one time. They can lead or co-lead 2 seminars and together with local students and local researchers they will create a critical mass to achieve significant results. 46 person/months of visits in other state institutions will provide each task a possibility of two 1-semester (3-6 months) visits. Due to financial limits, these plans are very modest, taking into account the thematic width and multidisciplinarity of the  SPADE 2. However, we want them to be accompanied by other national and international projects. These include Polish grants by Ministry of Science and Foundation for Polish Sciences. On the international level we have been applying in Marie Curie competition (HRM), for more details see B6. Also the Banach Center, which has local financial support, will be used for this. After 2 years we shall write a midterm report, in which leaders of each task, that will have had already 1 semester of intensive activity, will summarise the achievements and prepare a precise plan for the second 2 year period. Recruitment of experienced researchers will be conducted via the ToK web page, with the deadline preceding the semester of the activity by half a year. The recruitment will be executed by the Recruitment Committee, see B4.1. 

The ToK at the host institution will be provided via:
•	Traditional weekly (or bi-weekly) seminars, general and specific.  There are active seminars at IMPAN in all the areas of mathematics involved. They will be significantly strengthened by visiting researchers. This is already occurring thanks to visitors of the IMPAN-BC Centre of Excellence, which BANACH SPADE will extend. Mathematicians from universities in other cities (such as Kraków, Katowice, Gdańsk, Toruń, Lublin, etc.) participate in such seminars. 
•	Special 1-semester seminars, related to visits of foreign experts.
•	Weekend meetings and Friday-Saturday seminars
•	Semester (or shorter) special courses
•	Working groups
•	Individual tutoring
Visits abroad will be scheduled prior to semesters of intense activity in each task, so that the acquisition of new knowledge prepares ground for the successful transfer and development of it in the next stage of the programme.

B2. Transfer of Knowledge Activities
B2.1. Need for training and transfer of knowledge
For many years Polish science has been suffering from significant under-funding and the resulting brain drain of smart young researchers, mainly to the USA. Consequently there is a drastic need to remedy this situation. This can be done among others by a better participation of Polish scientists in the mainstream of research in several fields of great importance. Recent breakthroughs and new mathematical techniques, if adopted promptly in Poland, should increase the potential of Polish, and hence European, mathematics and its positive influence on the development of other fields.
A high level of knowledge in the analytic areas of mathematics usually correlates with the high level of technology and education in the engineering schools. We expect our ToK programme will have such an impact since PhD students and researchers of the Warsaw Technical University (which is located 200 m from IMPAN) are expected to participate in the activities of it. The topics of SPADE2 are selected not only to foster specialised research, but also to increase the mathematical analysis and stochastic processes knowledge/culture among engineers.
The project is of interdisciplinary character, so to foster technological development it is necessary to promote the interest of young researchers in this area. The topics of the project include very advanced, technically difficult theories. Even for experienced researchers it is important to learn this through direct contact with very top specialists who know the subject, its background, the motivations and relations to other subjects. It is also important to expose and discuss our own results with leading specialists and other younger colleagues.
Interest at the European level.
To be technologically competitive, Europe must organise and promote scientific activities in mathematics and applications, in particular in the SPADE2 areas. The Banach Center and IMPAN have a good location and perfect work conditions, including the main tool of a mathematician: the excellent library which has survived many economical and political turns. The Banach Center has long record of exchange of knowledge in mathematics but has little funds for medium term visits of external researchers, to attract them it should offer decent salaries (much higher then the standard salaries in Poland), not affordable within IMPAN budget, see below and B5,2. The scientific career is still in some esteem in Central-Eastern Europe and scientific potential is large attracting young candidates for research. Europe should not allow this to be lost!
Explanation of >4,>10, and "in", "out" mixture
The employment policy of IMPAN includes the rule that young (post-doc) researchers (coming from universities) are offered 1 - 2 year positions which are extended to longer periods only in exceptional cases. On the other hand IMPAN employs more experienced (post habilitation) researchers for longer contracts. Furthermore most of the knowledge we wish to import is on the cutting edge of sophisticated and advanced research. We thus plan to devote more person/months for visits and secondments of more experienced researchers.
We plan to invite 2 researchers for 1 year term and 12 for 2-6 months.
Planning 66 person/months visits of experienced and more experienced researchers during the programme, we expect usually 2 visitors from abroad at any time. They can lead or co-lead 2 seminars and together with local students and local researchers they will create a critical mass to achieve significant results. To increase the availability of the top experts in the field, we intend to offer contracts that would cover several months, but can be split up into parts.
 46 person/months of visits in partner institutions will provide ten short-term (2-6 months) visits.

B2.2. Content of the Transfer of Knowledge

IMPAN has organised seminars and lectures regularly for many years. They attract an audience from the whole of Poland with many talks given by foreign specialists. We would like to link each of the lectures of the proposed specialist with an established seminar. We hope that this will ensure a large audience for both seminar and lecture. Some seminars will be reorganised, to fit better the needs of the programme. We list existing seminars, which will be hopefully reinforced, due to the programme. In square brackets we indicate the leaders of the seminars.

1. General seminars:
•	Population dynamics (all the Tasks, Katowice/Warsaw) [R.Rudnicki, A. Lasota] 
•	Dynamical systems (Task 1, in relation with 2,3)[H. Żołądek, A. Zdunik, F. Przytycki]
•	Partial Differential Equations (Task 2) [B. Bojarski, W. Zajączkowski, G. Łysik, G.  Łukaszewicz, Z. Peradzyński, J. Grabowski]
•	Stochastic Processes (Task 3) [J. Zabczyk, Sz. Peszat, T. Komorowski]
•	Functional Analysis (Task 4) [A. Pełczyński, Cz. Bessaga, W. Żelazko, S. Rolewicz, S. Kwapień, R. Latała and others]
•	Spectral Theory of Differential Operators (Tasks 2, 4, Kraków branch) [J. Janas]
•	Approximation Theory (Tasks 2,4, Gdańsk branch [Z. Ciesielski, A. Kamont, T. Figiel]
and several other seminars at Warsaw University and Warsaw University of Technology
2. Special interdisciplinary seminar:
We plan to organise a multidisciplinary weekly seminar being a series of colloquium talks in all the tasks of the programme. This will be coordinated with the traditional series of Colloquia by Banach Center, IMPAN and the Polish Mathematical Society.
3. Some specific seminars (existing or planned):
•	Analytic theory of differential equations (Task 2)[G. Łysik, H. Kołakowski]
•	Geometric Theory of Sobolev spaces (Task 4) [P. Hajłasz, P. Strzelecki, A. Kałamajska]
•	Wavelet bases in function spaces (Task 4) [P. Wojtaszczyk]
•	Navier-Stokes Equations (Task 2) [W. Zajączkowski]
•	Holomorphic iteration (Task 1) [J. Kotus, F. Przytycki, A. Zdunik]
Several (1-semester or longer) courses or series of advanced lectures given by the scientists involved in ToK will be provided. We mention only some of them:
•	Homoclinic tangency bifurcations (Task 1) 
•	Point processes and their applications (Task 3.) 
•	Evolution equations (all Tasks) 
•	Unconditionality of spline bases (Task 4)
Scientists involved in the ToK will give short courses and survey lectures at mathematical weekend seminars (national), Banach Center research schools, summer schools and workshops (international). 
4. Tutoring
Tutoring related to the courses will be delivered by the visiting scientists, to local young researchers (in order that the young participants and tutors master the topic).

B2.3. Expected impact on training and knowledge transfer

We hope that that transfer of knowledge between partner organisations, IMPAN, and the outside researchers participating in the programme will help to strengthen the capabilities of the involved researchers in very advanced mathematical methods and in some applications (population dynamics, fractals, blood motion). The aim is to strengthen interdisciplinary contacts. This should increase the potential of mathematics serving as a modern tool and language for other scientific disciplines. In the longer term this should have positive impact on technological development. 

The main aim is to strengthen the existing ones and create new areas of competence in the research and educational activities of the Institute in selected branches of mathematics linked together. Let us again list these newly emerged areas:
NSE asymptotic behaviour in time and space (non-regular domains) 3D-thin or axically symmetric (blood vessels). Geometric measure theory in multifractal analysis of attractors in NSE. Scaling limits in physical processes, see B1. They will be developed basing on our knowledge and experience in dynamical systems, theory of functions and stochastic processes.      

Long term visits of carefully selected specialists working in these subjects will have a direct impact, increasing our capacity for fruitful research and teaching mathematics, see B.5 and  B2.1. We plan to invite the strongest specialists in the world (some of them visited already 
Poland) J.-Ch. Yoccoz, J. Pisier, K. Gawedzki, A. Kupiainen, S. Smirnov.

 The Banach Center and IMPAN are known for their significant past contribution to the European mathematical community. We have to make a considerable effort to conserve this status when the political and economical situation of Poland is changing (joining the EU). This is one of the main purposes of the action of transfer of knowledge described in this proposal. We believe that thanks to such programmes IMPAN and its branch, the Banach Center, will become one of the leading mathematical institutions in Central Europe. This together with the further increase of European cooperation should prevent the transoceanic brain drain of talented young mathematicians. This will help some outstanding Polish mathematicians working abroad to make the decision to re-immigrate to Poland. 

We expect one more positive effect of the multidisciplinary character of the programme. Its realisation should strengthen the cooperation between local teams working in the areas of the programme. The lectures and seminars given by visiting specialists will gather specialists from different groups, discussing common problems. The programme of transfer of knowledge is o special importance for graduate students and young researchers. Direct contact with the leading specialists will help them to choose their subject of research and to establish international contacts. 

B2.4. Required involvement of experienced researchers

It is crucial in mathematical research to have access to the most recent techniques and results obtained by other research groups. The most effected transfer of knowledge is by direct contacts with the leading specialists. In order to acquire new techniques usually it is sufficient to have one or more relatively short visits when teaching duty is not required. Therefore we plan the typical secondment to last 2-6 months and typical visit in Warsaw to last 6-12 months. 
We are conscious that the large number of 5 partner institutions forces visits to some of them to be relatively short and allows only one visit. The partner institutions have been chosen by us as a result of thorough consideration and the choice follows from the multi-disciplinarity of the project. We have decided that in view of our needs and EC policies it is appropriate to concentrate more on incoming foreign scientists.
The experience of our Centre of Excellence (and Banach Center) shows that leading foreign scientists willingly visit us, due to the enjoyable environment found here, often for long stays.
However in many cases it is difficult for more experienced researchers to be on leave for a long time. So we plan to split up visits in this category into shorter periods.

Indicative breakdown:

	Hosted experienced researchers to be 
financed by the contract	
Total (a+b)

(c)	
	Experienced researchers 
(person-months)

(a)	More experienced researchers 
(person-months)
(b)		
1. IMPAN	30	36	66	
2. U. Warwick	4	10	14	
3. Paris 6	4	10	14	
4. SNS Pisa	2	6	8	
5. INRIA	4	0	4	
6. CAU Kiel	0	6	6	
TOTAL	44	68	112	

The proportion 30:36 visits at IMPAN  is a delicate consensus between desire to invite (under competition) top YR which is of our highest priority to create a young dynamical new school, and more experienced researchers necessary to teach the advanced areas of the ToK.  
B3.Quality/Capacity of the Host and Partners

B3.1. Profile of the participants
1. The host organisation. The Institute of Mathematics of the Polish Academy of Sciences (IMPAN) plays a central role in the Polish mathematical community. Founded in 1948 (since 1952 in PAN) to materialise a dream of leading Polish school mathematicians, IMPAN has been the focus of mathematical life in Poland. It publishes several international journals and serials: Acta Arithmetica, Annales Pol. Math., Applicationes Math., Banach Center Publ., Colloquium Math., Dissertationes Math., Fundamenta Mathematicae, Studia Mathematica. International collaboration has been lively due to the Stefan Banach International Mathematical Center (BC) founded 30 years ago, the international branch of the Institute. The international programme committee of BC consists of: Piotr Biler, Bogdan Bojarski, Sabir Gusein-Zade, Stefan Jackowski, Rolf Jeltsch, Jerzy Kaczorowski, Olli Martio, Marta Sanz Solé, Jan Slovák, Verá T. Sós,  and Łukasz Stettner.
The BC has organised numerous semesters, conferences, workshops and research groups offering a great opportunity for the knowledge exchange. Now IMPAN-BC hosts an FP5 Center of Excellence, Marie Curie Training Site (BANACH) in the areas of our ToK project and is a node of the Geometric Analysis RTN. Scientific staff consists of 83 mathematicians (~60 on full time positions), including 35 full professors (including 8 members of Polish Ac. Sci.) and about 20 1-year research positions for young researchers. Most of the staff works in Warsaw, but IMPAN also has small branches (2-8 people) in Kraków, Wrocław, Gdańsk, Łódź, Poznań, Toruń and Katowice (mainly at the math departments of universities). 
The following departments are involved in the SPADE2 project: the Department of Differential Equations, the Department of Functional Analysis the Department of Probability Theory.

Below we describe the state-of-the-art at IMPAN and cooperating institutions in (broad) areas including topics of SPADE2. YR in these groups will highly benefit from the SPADE2.

1. The Partial Differential Equations group at IMPAN is led by B. Bojarski and W. Zajączkowski and consists of 8 mathematicians. It organises an intercollegiate seminar on PDE's and two specialised seminars. Its research covers the study of equations of hydromechanics, Einstein equations, elliptic equations with connection to Fredholm operators and analysis on metric spaces, variational calculus and analytic theory of differential equations. The group strongly cooperates with mathematicians employed at Warsaw University (Faculty of Mathematics, Faculty of Physics, Interdisciplinary Center of Mathematical Modelling) and Warsaw University of Technology, Silesian University and Wrocław University, where the theory of nonlinear PDEs and attractors are strongly represented. The ToK programme should provide better understanding of asymptotic behaviour of solutions of semi-linear evolution equations, turbulence and regularity problem for the Navier-Stokes equations.
2. The dynamical systems group in Warsaw consists of about 20 researchers (including 5 PhD students). Most active topics: limit cycles for ODEs related to 16 Hilbert problem, 1-D real and complex dynamics and underlying fractal sets.  SPADE2 plans to develop the latter topic; a possible return (reintegration) from the USA of Grzegorz Świątek (invited speaker at the 1998 International Congress of Mathematicians) will foster it. We want to rebuild ergodic theory of smooth higher-dimensional dynamics in relation with classical mechanics and differential geometry, in particular due to  reintegration from the USA of M. Wojtkowski (invited speaker at the 2002 ICM). The population dynamics seminar in Katowice is becoming nationwide. We shall cooperate also with dynamical systems groups in Toruń (ergodic theory) and Kraków (computer assisted analysis of (quasi)periodic and chaotic motions), which will also benefit from the ToK. (Just last weekend we organised a meeting/school in dynamical systems in which 40 Polish students participated!)  
3. The stochastic processes group at IMPAN consists of 5 researchers. It cooperates with other such groups in Warsaw, Toruń, Kraków, Łódz, Lublin and Wrocław. It organises yearly a series (1 or 2 semesters) of lectures on advanced topics of probability for postgraduate students and young researchers. Its main scientific interest is concentrated on stochastic PDEs, stochastic control and filtering, mathematical finance, and stochastic fluid dynamics.  SPADE2 project will help us to develop new areas of stochastic analysis (stochastic equations on manifolds, superprocesses, Dirichlet form) with emphasis on application. This will make our group more interdisciplinary. 
4. The Function Theory and Functional Analysis team in Warsaw, lead by A. Pełczyński and S. Kwapień consists of more than two dozen mathematicians. The main subjects of research are: geometry of Banach spaces, especially function spaces, wavelets and approximations, local theory, probabilistic methods and operator theory. Our long term intention is to move the centre of gravity of our research from abstract geometric theory towards analysis and application. The SPADE2 project opens a very good opportunity to support this plan, attributing interdisciplinary character to our research. 

Key scientific staff involved in the ToK project (all 50% involvement) is listed below, with some areas of expertise
(for the complete list of this scientific staff involved see brackets in B1.4). 
Task 1: Prof. Feliks Przytycki  --  holomorphic iteration, (non)-uniform dynamics,  
Dr Michal Rams – non-conformal iterated function systems, 
Prof. Maciej Wojtkowski – billiards, non-uniform hyperbolicity in mechanics and geometry. 
Task 2: Prof. Wojciech Zajączkowski PDEs, Equations of viscous fluids.
Dr hab. Grzegorz Lukaszewicz – Attractors in Navier-Stokes Eq.
Dr hab. Grzegorz Łysik – Evolution PDEs, asymptotics 
Prof. Ryszard Rudnicki – Stochastic processes, population dynamics,
Task 3: Prof. Jerzy Zabczyk (member of Polish Ac. Sci.) – Stochastic Processes
Dr hab. Szymon Peszat – Stochastic Processes
Dr hab. Tomasz Komorowski – Turbulent transport, 
Task 4:  Prof. Aleksander Pełczyński (member of Polish Ac. Sci.) Functional Analysis, 
Dr hab. Michał Wojciechowski – Function spaces,
Prof. Przemysław Wojtaszczyk – Wavelets methods

The number of mathematicians in Warsaw active in the domains covered by our ToK project easily exceeds 50. They are employed mainly in Math. Dept. of Warsaw University and the Technical University of Warsaw. IMPAN has signed an agreement with these institutions about a joint PhD studies programme, which puts a number of PhD students involved in the tasks of the ToK project above 50. Also numerous undergraduates (4-5 year of studies) will greatly benefit. Also researchers from other Polish cities with easy access by train will be involved; in particular researchers from other branches of IMPAN. Our partner, Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) of Warsaw University specialises in theory of complex structural phenomena, biophysics, biomedicine, atmospheric modelling and large scale computing. It participates in four FP5 programmes (in cooperation with the IWR – Heidelberg)

PARTNERS

Each Partner Institution of the Project has been selected by IMPAN taking into account the following.
•	Recent substantial breakthroughs/advances achieved there, which we would like to implement at IMPAN/Poland. The institution top scientific level. Representing expertise  in several tasks of the project
•	Traditions in the subject already existing at IMPAN (or broader: in Warsaw/Poland) and existing contacts with the Partner.
•	Broader needs to develop those areas of mathematics which are applicable in modelling and analysis of deterministic and stochastic processes as described in B1 and therefore to increase Human Resources of Warsaw/Poland/this part of Europe to cope with the needs of the challenges of the technological development. 
These are partners for the outcoming visits, to be financed by the ToK. In fact we will collaborate and visit more partners, but these visits will be financed from other sources, see B6. We plan to use most of money in SPADE2 for incoming visits to pay decent salaries, see B2.1 and B5,2.

2. The University of Warwick, Mathematics Institute (U. Warwick) - D. Elworthy. Founded in 1965 by Professor Sir Christopher Zeeman, Warwick has one of the top UK math institutes. It hosts the Mathematics Research Center (MRC) (current director D. Elworthy) which plans and organises the visitors' programme. Run since 1965, the year-long research symposia attract mathematicians of international stature, several of whom are related to our ToK programme. The Institute now hosts 3 Marie Curie FP5 Training Sites: in Algebraic Geometry (3-fAG), Dynamical Systems (DYWA) and Stochastic Analysis (SETSAC), and coordinates the ESF programme Probability Methods in Non-hyperbolic Dynamics (of which Warsaw is one node, with coordinator F. Przytycki). In dynamical systems researchers concentrate on 1-dimensional real and complex dynamics (W. Shen, A. Epstein, O. Kozlovski, S. van Strien, A. Manning, A. Epstein), ergodic theory (P. Walters, W. Parry), Kleinian groups (C. Series, D. B. A. Epstein, V. Markovic), applied dynamical systems (Rand, MacKay, Barkley, Stuart, Stewart, Baesens, Robinson and many others), Hamiltonian dynamics (Gelfreich, MacKay).
3. Universite Pierre et Marie Curie (Paris 6) - G. Pisier. Equipe d'Analyse Fonctionnelle of the University Paris 6 is one of the leading centres in France and Europe of functional analysis. The main directions of its research are: algebras and operator spaces, Banach space geometry, potential theory. Among the members of the faculty there are 8 professors, and there are  4 other professors in the associated unit of CNRS (G. Pisier, A. Volberg, S. Szarek, G. Godefroy, M. Tallagrand). They organise 3 regular weekly seminars: Functional Analysis (founded by L. Schwartz), Introduction to Analysis and Potential Theory. They play a central role in their subjects in France and they attract mathematicians from all around the world. In Paris 6 there is also the Laboratoire de Probabilite et modeles aleatoires (J.-P. Thouvenot) which cooperates with Paris 7 ( A. Chenciner, V. Baladi), Task 3. 
4. Scuola Normale Superiore Di Pisa (SNS Pisa), - G. Da Prato. Scuola Normale Superiore Di Pisa is one of the leading Italian centre of education and research. The department of Mathematics consists of 14 permanent researchers and a number of PhD students, postdocs and visitors. It is a very strong centre of analysis and geometry, in particular in variational calculus (L. Ambrosio), PDEs, SPDEs, and control theory (G. Da Prato), complex analysis (G. Tomassini), and differential geometry (F. Ricci). In 2001, the Research Center Ennio De Giogri was established in order to gather Italian and foreign researchers with the idea of organising semesters focusing on a research area of particular relevance. 
5. Institut National de Recherche en Informatique et en Automatique (INRIA Rocquencourt) - J. Lévy Véhel. INRIA is big multisite organisation (1900 scientists). Its Rocquencourt Research Unit (accessible by RER from PARIS) includes 36 teams (520 people). The team has numerous industrial relations. We plan our ToK mainly with "Projet Fractales" team (Fractals, complex models and artificial evolution) in the topics:
1. Fractal Analysis and Modelling: multifractal analysis, 2-microlocal analysis, fractal stochastic processes. 
2. Evolutionary Algorithms. 
6. Christian-Albrecht-Universitaet zu Kiel (CAU) – H. Koenig. At the Mathematisches Seminar of CAU there is strong analysis group (H. Koenig, D. Mueller, W. Bergweiler). Among research topics represented in the faculty there are: Functional Analysis (geometry of Banach spaces, operator ideals, local theory of Banach spaces and convex geometry), Harmonic Analysis and PDEs (Euclidean Fourier Analysis, Analysis on Lie Groups and Applications to PDEs), Complex analysis (complex dynamics, value distribution differential equations in the complex domain).

B3.2. Quality of infrastructure of the host organisation and partners 

Library
The library of the Institute is the Central Mathematical Library of Poland. The collection of books (above 70,000) and journals (more than 630 titles) places the library among the best mathematical libraries in the world. The library collects scientific publications (in congressional languages and in Polish) concerning mathematics and its applications. Since 1993 a database of acquisitions has been accessible via the Internet. The catalogue of books and journals is available on our Internet homepage: http://www.impan.gov.pl/LIB/. The library is open to the general public (weekdays 8:00-19:00, Saturdays 8:30-12:30); students and researchers not working at IMPAN are also allowed to borrow books. Xerox facilities are also available. The library cooperates with many mathematical departments all over the world, exchanging publications and information.

Computer facilities
All of the computers at the IMPAN are connected to a local network. They are Pentiums, PC486, Sun Ultra 5 servers and SunSPARC workstations.
The computers are connected to the Internet (domain: impan.gov.pl). The main servers work under the Solaris operating system, others under Windows, Linux and DOS. Mathematicians can use Mathematica, Maple, Matlab, C++, and some GNU software.

Dissemination of knowledge
The Publication Department of IMPAN publishes 8 internationally recognised journals and serials (see B3.1). Most of the serials have international editorial committees. In particular, small monographs can be publish quickly in the Banach Center Publications. Also preprints of IMPAN give the possibility to spread instantly new knowledge. Since our journals and preprints are available on the Internet (www.journals.impan.gov.pl) there is no problem with world wide access. The programmes of seminars and abstracts of lectures can be also found on our web site (www.impan.gov.pl/biuletyn).

Accommodation
IMPAN holds guest rooms for 20 visitors (there will be 10 more in the near future). The guests are also offered offices with computer facilities.

Advertising
The SPADE2 project will be advertised on the web page of IMPAN as well as in the European Newsletter, where the vacancies for experienced and more experienced researchers will be published. We shall also send posters to the main universities in Europe. Local seminars and lectures are advertised by posters, web page and e-mails sent to interested participants. The ToK successes and events will be advertised on CORDIS web page, too.
A bulletin of  SPADE2 is planned to be issued and sent to the participants regularly.

Partners
All partner organisations have sufficiently good infrastructure and facilities necessary to conduct mathematical research. They all have libraries, excellent internet connections and computer facilities.

B4. Management and Feasibility
B4.1. Proposed management and organisational structure
The organisational and management structure of the project is illustrated on the following chart:

Partners
Scientists-in-charge

International Advisory Board

Director of IMPAN

Scientific 
Committee

Coordinator
of the project

Administration

Recruitment
Committee

Task 
Coordinators

The key management is played by the coordinator who directly communicates with the Director of IMPAN and EU institutions.
1. The scientific committee (which includes Tasks Coordinators and Recruitment Committee) decides about general priorities, the choice of topics and methods to be transferred and the general splitting of means among the tasks. It suggests to the coordinator and task coordinators the general methods of organising the knowledge transfer (e.g. a common seminar). It evaluates the yearly progress in the top programme.
2. The coordinator is responsible for the execution of the ToK programme. He collaborates with the Scientific Committee (in particular with the task coordinators) in: 
(a)	selecting the candidates in IMPAN for the secondments, in the cooperation with the partners scientists-in-charge)the place and period of their visits; 
(b)	organising the transfer of methods learned during the secondments after return to Poland;
(c)	selecting the subject and time and advertising the visiting position at IMPAN in the ToK programme, with the help of  the technical staff. 
The coordinator is responsible for financial decisions made in the ToK programme and takes care of all administrative and financial matters (offices, accommodation) using the administration structure of IMPAN. He also collaborates in these matters with partner organisations
3. The Tasks 1 - 4 are coordinated by the task coordinators chosen among scientists involved in the project. They are subordinated to the coordinator of the project. They organise visits (timetable, selection of topics) communicating with scientists-in-charge from the partner organisations. On the IMPAN side they fix the content and timetable of seminars, lectures and workshops connected with the ToK project. 
4. The International Advisory Board, IAB (consisting of distinguished scientists) will be established to ensure the maximal possible efficiency and help to indicate the best experts possibly involved in the transfer of knowledge. 
5. The Recruitment Committee will select YR candidates (see B3.2 Advertising). 
It demands recommendation letters and asks experts for the opinions. 

More experienced researchers (experts) will be selected carefully by the Scientific Committee
upon suggestions by tasks coordinators  (and advised by IAB) basically from the list  in B1.4
square brackets and will get personal invitations.

The execution of the Project will be monitored and evaluated yearly by the Scientific Committee in collaboration with the partner scientists-in-charge. The progress (measured e.g. by publications, attracted students etc.) will be reported to the EU commission by the Coordinator.
Very good communication with the Partner Institutions is expected on the base of excellent existing contacts. We plan a part time employment of a technical secretary in charge of the project, in particular in charge of technical preparation on an e-bulletin, and creating and maintaining the web page by a professional.

B4.2. Management know-how and experience of the host

IMPAN has big experience in organising scientific events such as Conferences, Workshops and Semesters (over 20 events yearly). IMPAN was the main organiser of the International Congress of Mathematicians in 1983. Yearly we host a few hundred guests from abroad offering them excellent conditions of work and accommodation. We own guest rooms for 20 visitors and there will be 10 more in near future.
We organise over 20 regular seminars, many of them with a very long tradition.
The annual budget of IMPAN exceeds 3 million Euro. However, only 50% of this amount is obtained directly from Ministry of Science, former  Office for Scientific Research (KBN) (the remaining funds have to be gathered from other sources). The scientific staff exceeds 70 people. Apart from the main centre in Warsaw we have 7 branches in other cities and a modern conference centre in Będlewo (with facilities for mid sized conference).
We have very well developed international cooperation. Let us mention again our IMPAN-BC Centre of Excellence and Marie Curie Training Site: Banach Center for Differential Equations, Functional Analysis and Stochastic Processes, and else our cooperation with the FP5 networks: Marie Curie Control Training Site (coord.: SUPELEC - Gif-sur-Yvette) and Geometric Analysis RTN (coord. N. Teleman - Ancona), see also B6.

B5. Relevance to the Objectives of the Scheme.

1. New areas of competence. 

The objective of developing new areas of competence has been addressed in B1.2. We listed there a large spectrum of research objectives in which we hope to achieve a substantial scientific progress. These are all interlinked, see B1.1 (or B1.4.motives for example). Analytic-stochastic areas are adequate to provide a strong stimulus to the development of applications of mathematics in technology.

The multidisciplinary profile of SPADE2 which includes the prominent areas of deterministic and stochastic processes, chaotic phenomena and fractal structures, treated in a rigorous mathematical way, will be a magnet for young researchers from Warsaw, Poland and the region.

We made a choice of 5 partner organisations where we plan to send our researchers to acquire knowledge (we plan stays from 2 to 4 months; see B2.4). After returning to Poland these researchers will give new directions to the traditional seminars, grouping local scientists (listed in B2.2), and organise and lead new seminars and courses. 

2. Availability of appropriate researchers.

An important role will be played by incoming top-class researchers from several European institutions and also from outside of Europe. There is a strong tendency to invite very experienced researchers. However to build our scientific centre, we want to do our best to attract younger (less then 10 years of experience) researchers, who are at the forefront of 
their subject to share with us their enthusiasm. 

Adequate salaries and other financial contributions offered by the programme will attract foreign researchers for long stays. We do not doubt the availability of appropriate researchers willing to come to transfer the requested knowledge. We have strong international links and many scientific papers and books have been written by the employees of our institute in co-autorships with top foreign scientists. A great role has been played by our IMPAN-BC 5FP Centre of Excellence which has been visited for at least 1 month stays by many top scientists. Traditions of Banach Center semesters, workshops and other initiatives also are very important.

3. Local critical mass of researchers.

The local research groups are described in B4.2. We give below an additional information how these groups are involved in training of young researchers and PhD students. The 5FP programmes (our Marie Curie Training Site BANACH and CE IMPAN-BC) and hopefully the ToK we apply for, have been making a big pressure to increase substantially the number of PhD and postdoc positions in place of professors becoming retired. There are numerous very talented PhD and undergraduate students available and we want to absorb them in science (see also B6). A substantial increase of the number of such positions is planned already for 2003/2004, despite financial difficulties. The ToK and EST (for which we shall apply in November) will create pressure to increase financing of the Polish YR.
Therefore the ToK would fall on very fertile ground. We have recently signed agreements with the Department of Mathematics of Warsaw University (3 years ago), and the Department of Mathematics and Computer Science of the Warsaw University of Technology (last year) on joint PhD studies programmes. Our PhD students lead exercises/tutorials with undergraduates there. 

Moreover most of the scientists - authors of the proposal (including the coordinator), teach advanced courses and lead seminars at Warsaw University, Warsaw University of Technology and Silesian University attracting the best students.

Recently our links with the Warsaw University of Technology increased. This promotes cooperation with technologically oriented researchers. We are in the course of founding a new Mathematical Centre for Science and Technology, within the IMPAN (to be launched in the Autumn 2003). IMPAN seminars and courses attract advanced undergraduate students and YR from all the universities in Warsaw and also from other cities (Katowice and Kraków are 2.5 hours away by train, Łódź is even closer, and participants come also from Lublin, Gdańsk, Toruń etc ). We organise a weekly colloquium for PhD students, inviting top senior professors and the best YR to deliver lectures. 

4. International dimension.

Although the Banach Center has been acting for 30 years, it has never been in a position to offer mathematicians from Europe and outside medium term positions (semester or longer) with decent salaries. Therefore, this part of our international activity was not developed to the desired degree. The ToK project opens the possibility to fill this gap.

We plan to transform IMPAN into international research and training institution, using all the existing resources and applying for additional ones, in cooperation with local Universities and international organisations. We hope that it becomes a leading mathematical institution of this type in the central-eastern Europe.

The ToK will strengthen the international dimension of IMPAN. Together with the national PhD studies it will attract PhD students from abroad, with fellowships from other sources. Already in 1995-99 we organised special international PhD studies at IMPAN, but that programme has been suspended because of the lack of adequate financial support. We would like to revitalise these studies. Recently we received financial support from the Polish Committee for Scientific Research (KBN) for PhD studies with the participation of foreign participants. EST (see above) would also play a crucial role.

B6. Added value to the community

A central issue of every modern society is solving problems which appear in technology, ecology, economy and medicine. Mathematical models help expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. The increased diversity of scientific problems poses new challenges to mathematics.
Many of these ideas were invented and developed in Europe and have now been studied world-wide. It is quite possible to maintain European lead in these fields, and this project would be an important contribution to that aim. Having a strong mathematical centre in Warsaw, an important scientific centre in Eastern/Central Europe with a big well educated population, will highly contribute to the European Research Area.  

Extension/reinforcement of present activities towards scientific and technological development of the region.

IMPAN continues great traditions of Polish mathematics and the project concerns some of these traditional topics on which bases new areas of competence will be built. It is necessary to transfer to its investigations new mathematical ideas and techniques and to open them to applications. Already in FP5 we have undertaken efforts to develop, on the basis of the traditional areas, modern mathematical disciplines which lead to applications substantial for technological development. The EC programme of the Centre of Excellence IMPAN-BC has been created. However it will definitely finish on  30 September, 2004. 
IMPAN-BC included the packages:
1. Public-Key Cryptography and Computational Number Theory
2. Information Theory and its Applications to Physics, Finance and Biology
3. Mathematical Modelling and Analysis of Cellular Populations
4. Nonlinear Systems and Control
5. Mathematics of Finance and Stochastic Control
6. Approximation structures in Banach spaces
7. Symplectic Singularities and Applications
8. Visual Modelling

In SPADE2 we plan to develop new areas of competence, growing from some subtopics in the above packages, including new topics, but better interrelated and much more focused.  
The topics of SPADE2 also continue some topics of the FP5 Marie Curie Training Site at IMPAN named Banach Center for Differential Equations, Functional Analysis and Stochastic Processes (acronym: BANACH).
The topics are related to the topics of the FP6 RTN CODY and CODY2 (Conformal Structures and Dynamics), proposals submitted by IMPAN, which got very high notes but have not win.
The Polish Office for Scientific Research (KBN), which finances IMPAN and the Banach Center (in more than 50%), is providing additional money for individual (or small team) grants. There are now several such grants at IMPAN, some of them in the areas of the ToK project (4 right now). Also Foundation for Polish Science (FNP) provides some means (for example FNP subsydium supported in summer 2003 participation of our four students in CIME School in Turbulence in Martina Franca. KBN has also been providing additional money for EC programmes, in particular the Centre of Excellence and Marie Curie Training Site, and it is probable that KBN will continue this policy with respect to FP6. These local means, when added to EC financing the ToK project, should allow us to contribute significantly to the training of Polish participants. It should also allow us to increase the number of local PhD and postdoc fellowships, finance participation in conferences and seminars, etc.
At present we have a demographical peak in Poland with numerous very gifted students in mathematics and we feel responsible for their scientific careers. Without adequate support they will be lost to science or they will emigrate (we have high unemployment rate). A pressure is necessary to increase Polish poor 0.4 percent of GDP spending on research.
Finally the ToK (combined with other programmes) will attract young researchers (YR) from neighbouring and NIS states and foster reintegration of experienced researchers from the USA (this process has already started).
We hope that due to contacts with the ICTP in Trieste also contacts with YR from Asia, Africa and Latin America will develop. Dynamical systems group has already contacts (joint papers) with people at IMPA (Brasil) and Chile. We plan to cooperate with other EC programmes active in the areas of SPADE2,  in particular with FP5 HYKE and HARP RTN’s, (nodes in Wroclaw), 6FP RTN Modelling of tumor growth and therapy (node at Warsaw University) and PHD (Phenomena in High Dimensions) node at IMPAN (invited for negotiations). 
Anticipating FP6 our IMPAN-BC Centre of Excellence signed a cooperation agreement with Centres of Excellence in math: EURROMMAT, MATHIAS, BIS-31 in Romania, Hungary and Bulgaria, and some other institutions. If our SPADE2 is accepted we plan to involve these centres, thus further contribute to the extension of ERA.
 
Attractiveness of Europe means, in particular, high level research centres open to foreign visitors and attracting outside researchers by their competence. It also means presence of authorities in science, in particular in mathematics and its applications. Our ToK programme aims, in particular, at putting at higher level the research and researchers in the domain of SPADE2 in Poland and outside. Especially important aim is to attract to science and to rise research qualifications of young people. The coming 10 years in Poland are crucial for the recruitment of candidates for research for two reasons: (a) the population wave at the universities attains its maximum, and (b) the unemployment rate is high (18%). Both factors improve decisively the chances of recruitment of young scientists. The ToK programme should contribute considerably not only to research level of the SPADE2 group and related scientists, but also shows to young graduates that research career in Poland has positive perspectives. 
Clearly, the possibility of transferring knowledge, offered to the outside researchers – experts in the domain of SPADE2 – increases attractiveness of the Polish research centres. Indirectly, the rising of expertise in the applied oriented research SPADE will increase the level of mathematics taught at universities. (see B2.1)                                                  

Gender Issues. At present the rate of women employed at research positions at IMPAN is about 20%  and 6 out of 23 in the subjects of the SPADE2 project. These proportions are likely to become more balanced when the young participants (postdocs and PhD students) will become involved in the project. We will pay a special attention to this issue and make efforts to achieve better balance.

B7. Previous Proposals and Contracts

This proposal is not a resubmission, however it has some common points with the proposal No FP6-509757 BANACH SPADE in FP6-2002-Mobility 3, deadline 22 May 2003, which failed.
The present proposal, while remaining multidisciplinary, is much smaller and focused.
B8. Other Issues

Does the research presented in this proposal raise sensitive ethical questions related to: 	YES	NO	
•	Human beings		√	
•	Human biological samples 		√	
•	Personal data (whether identified by name or not)		√	
•	Genetic information		√	
•	Animals		√	

We confirm that the research presented in this proposal does not involve:
•	Research activity aimed at human cloning for reproductive purposes,
•	Research activity intended to modify the genetic heritage of human beings which could make such changes heritable;
•	Research activity intended to create human embryos solely for the purpose of research or for the purpose of stem cell procurement, including by means of somatic cell nuclear transfer;
•	Research involving the use of human embryos or embryonic stem cells with the exception of banked or isolated human embryonic stem cells in culture.

ENDPAGE

HUMAN RESOURCES AND MOBILITY (HRM)
ACTIVITIES

MARIE CURIE ACTIONS

Marie Curie Host Fellowships for the Transfer of Knowledge (ToK)
Development Host Scheme

PART B

BANACH SPADE
♠♠

DIFFERENTIAL EQUATIONS, FUNCTIONAL ANALYSIS AND
STOCHASTIC PROCESSES