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 GuseinZade, 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 IMPANBC 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 1year research positions for young researchers. Most of the staff works in Warsaw, but IMPAN also has small branches (28 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 stateoftheart 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 semilinear evolution equations, turbulence and regularity problem for the NavierStokes 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, 1D 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 higherdimensional 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 – nonconformal iterated function systems,
Prof. Maciej Wojtkowski – billiards, nonuniform hyperbolicity in mechanics and geometry.
Task 2: Prof. Wojciech Zajączkowski PDEs, Equations of viscous fluids.
Dr hab. Grzegorz Lukaszewicz – Attractors in NavierStokes 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 (45 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 yearlong 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 (3fAG), Dynamical Systems (DYWA) and Stochastic Analysis (SETSAC), and coordinates the ESF programme Probability Methods in Nonhyperbolic Dynamics (of which Warsaw is one node, with coordinator F. Przytycki). In dynamical systems researchers concentrate on 1dimensional 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, 2microlocal analysis, fractal stochastic processes.
2. Evolutionary Algorithms.
6. ChristianAlbrechtUniversitaet 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).
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