Project
Objectives
The overall aims are to understand local self-similar structure of fractal spaces, objects or processes, by methods of space-time conformal rescaling. The investigations and the subject of the training programme of the proposed network will therefore be contained in three closely related fields of mathematics and mathematical physics whose common denominator is the notion of conformal or quasiconformal structure: mathematical (conformal) analysis and geometric measure theory, conformal and low-dimensional dynamics, continuum scaling limits of physical processes.
The following are objectives for the network:
**A. Conformal Analysis and Geometric Measure Theory **
A1. Conformal structures, analytic and geometric background and view, Objective: A systematic study of mappings with finite distortion. quasiconformal mappings in metric spaces, applications to limit sets in dynamics and ideal boundaries.
A2. Potential theory, analytic tools. Objective: Description of analytically and quasiconformally removable sets, convergence of singular integrals and rectifiability. Applications to Julia sets and rigidity in holomorphic dynamics. Investigations towards the Brennan conjecture.
A3. Topics in Fractals and Multifractal Analysis (see also A2, P1-2 and themes D). Objectives: Construct a general theory, involving projection and intersection schemes. Derive a comprehensive multifractal description of new classes of measures emerging from deterministic processes.
**D. Conformal Dynamical Systems **
D1. Iteration of interval and circle maps, and their complexification, weak hyperbolicity and physical measures. Objective: The real Fatou conjecture and the Thom-Smale-Palis objective.
D2. Geometry of dynamical and parameter space. Objective: Improve the understanding of the rich geometry of both dynamical and parameter spaces, including the interplay between the two, for various families of rational or holomorphic maps in one complex variable.
D3. Hausdorff measure and dimensions. Objectives: Determine the conformal measures for various classes of Julia sets. Understand the periodic orbits and dynamical Zeta-function, compare P2.
D4. Limit sets for Kleinian groups and relations. Objectives: Understanding conformal measures supported on "deep" points. Find restrictions on rational maps via equivalence relations, more subtle than affine laminations, being a counterpart for the extension of the action of a Kleinian group to the bal.
D5. Beyond dimension 1. Objectives: Prove a "no wandering domain" theorem for HÈnon mappings; Find a decomposition of the dynamical space analogous to the Yoccoz puzzle. Explore the parameter space for the famous HÈnon maps corresponding to non-uniformly hyperbolic analogues of solenoids.
D6. Iterated Function Systems (IFS). Objectives: Improve the understanding of IFS with overlap and the Hausdorff and packing measures of limit sets (see A3). Determine the dimension of sets of parameters with defected dimension of the limit set or singular probability distribution in the "fat" case. Study infinte IFS’s (arising in renormalization techniques D1,D2).
**P. Topics in Mathematical Physics (see also A3)**
P1. Scaling limits in physical processes. Objectives: Build a bridge between the probabilistic approach to random growth processes and conformal field theory in fixed and in fluctuating geometry. We expect progress in rigorous foundations for renormalization and universality for 2D critical lattice models. Extend research in other random models, where complex analysis plays an important role: Diffusion Limited Aggregation (a generic model of fractal growth), random matrices (of major importance in studying disordered media), etc. Make progress in the study of Schramm-Loewner Evolution and various lattice models (Ising, dimer models, SARW).
P2. Infinite dimensional systems. Objective: Study coupled map lattices and more general infinite dimensional systems
P3. Turbulent transport. Objective: Explore IFS approximations not only for modelling passive transport in synthetic turbulence but also for other transport phenomena of practical importance: high Reynolds number flow and porous medium flow in multiscale materials. |