## IV Spring School in Analysis to the memory of Aleksander Pełczyński

### Programme

The programme consistes of three courses:

1) Sergiei Kislyakov (St. Petersburg) "BMO-regular lattices of measurable functions and applications"

The lectures are intended to review some results (including some recent ones) and techniques pertaining to the theory of interpolation of Hardy-type subspaces in lattices of measurable functions and their ramifications and applications. The notion of BMO- regular lattice introduced by N. Kalton plays a crucial role in all that. Among other things, the relationship with certain problems of the classical Fourier analysis, certain versions of the Grothendieck inequality, the Carleson corona theorem, and a fixed-point theorem for multivalued maps will be exposed.

2) Andrzej Żuk (Paris) "Analysis and Geometry on Groups"

• Burnside's problem, finitely generated but infinite groups with all elements of finite rank,
• Milnor's problem: groups of growth slower than expotential but faster than polynomial,
• Day's problem: exotic groups with a Banach mean,
• Atiyah problem: manifolds with irrational L² Betti numbers,
• The construction of Margulis: families of strongly connected graphs,
• Problem of Gromov: groups of expotential but non-uniform growth

Simple solutions of above problem will be presented. We will also talk about Groups generated by automata, property (T), random groups and graphs, L² invariants of groups and manifolds, random walks on groups and graphs.

3) Błażej Wróbel (Wrocław/ Piza) "Spectral multipliers for generators of symmetric contraction semigroups"

We demontrate various types of multiplier theorems for generators of symmetric contraction semigroups. A multiplier theorem allows one to obtain a well behaved functional calculus for certain classes of multiplier functions. We mostly work on $L^p,$ $1<p<\infty.$. The focus of the lecture lies in proving general multiplier theorems for single operator or systems of commuting operators. If time permits, we will also give some applications to the $L^p$ biundedness of various Riesz transforms.

There will be four 1,5- hour talks daily and many possibilities for discussions.