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 stateoftheart and contribution to the advance of capabilities of the applicant, by research fields.
1. Dynamical systems. Many investigations have been encompassed by the general ThomSmalePalis 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 nonconformal 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 (SinaiRuelleBowen) 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 NavierStokes 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 NSflows 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 resummation of formal solutions and apply it to nonlinear evolution equations such as semilinear 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 reactiondiffusion 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 subRiemannian 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, superprocesses 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 twodimensional random growth models and random matrix theory. Carleson and Makarov provided a deterministic model, via Loewner Equation, of diffusion limited aggregation (DLA) (WittenSanders 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 nonlinear approximation – collaboration with A. Shadrin (Cambridge). The connections between the measure theory, nonlinear 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 (nonreflexive 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 1D 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. RiveraLetelier Antofagasta] ;

Hyperbolic and dominated splitting, nonuniform hyperbolicity: Lorentz and Henon attractors, SinaiRuelleBowen measures, homoclinic tangency bifurcations, limit sets for nonconformal 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 quasiperiodic 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, SRBmeasures, 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 NavierStokes [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 SchrammLoewner evolution, random matrices [W. Werner  ParisSud, 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 [JL. 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, CarnotCaratheodory spaces [G. Pisier – Paris 6, GarciaCuerva – 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, nonlinear 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 colead 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 1semester (36 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 biweekly) 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 IMPANBC 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 1semester seminars, related to visits of foreign experts.

Weekend meetings and FridaySaturday seminars

Semester (or shorter) special courses
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
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