The subject of SPADE2 is interdisciplinary: methods of dynamical systems, 1,2, finite and infinite dimensional, the study of regularities and long time behaviour of solutions for some PDE's, as Navier-Stokes Eq., and fractal nature of arising attractors.Applications include fractals (data compression, signal transmission), flows in blood vessels, population dynamics, tomography. More specifically, areas and methods include:
- 1D-dynamics, real and complex, Iterated Function Systems (IFS) as a kernel of higher dimension phenomena (hyperbolic, non-uniformly hyperbolic, dominated splitting), dimensions and measures on metric attractors and basins, Multifractal Spectra of Lyapunov exponents, entropy, dimensions.
- Infinite-dimensional systems. 2D (Navier-Stokes Eq.), 3D thin domains attractors
- Scaling limits in physical processes. Dimensions and scaling exponents for the 2D models like percolation, Ising model, self avoiding random walk, etc. There are expected advances in the models like diffusion limited aggregation (a generic model of fractal growth) or random matrices (of major importance in studying disordered media).
- Stochastic Processes. Evolution equations with impulsive noise. Stochastic Burgers and Navier Stokes equations. Stochastic equations for spin systems and bond markets. Bellman equations and stochastic maximum principle for controlled evolution equations. Random environment and turbulence. Stochastic equations on manifolds.
- Methods of Function Spaces: Sobolev Spaces, Gerrey classes functions of bounded variation, analysis of singularities of measures, singular integrals and harmonic analysis. Wavelet methods. (Degenerate) quasiconformal methods in holomorphic dynamics and PDE's.
The corresponding tasks are topics in
- Dynamical Systems, Fractals
- Partial Differential Equations, turbulence, asymptotics
- Stochastic Processes, scaling limits.
- Methods of Function Spaces.
The priorities in the project are the visits of top specialists from abroad giving series of lectures, and seminars with the participation of researchers and PhD students from IM PAN and other institutions. A special interdisciplinary ToK SPADE2 weekly seminar will be organized.
The partners of IM PAN for outgoing visits are the universities: Warwick, Paris 6, CAU Kiel, and research institutes SNS Pisa, INRIA Institute at Rocquencourt ("Projet Fractale").
The activities of ToK will be widely advertised through Web pages with abstracts of lectures and seminars, bulletins e-mailed to registered participants and PhD students. Some ToK survey lectures and courses will be published in Banach Center Publications. Openings of positions and vacancies will be published on the web page and adequate European bulletins.