### Invited lectures at 50th Winter School

**Alan Chang:** The Kakeya needle problem

The Kakeya needle problem asks the following question. What is the area of the smallest region in the plane in which you can rotate a unit line segment 180 degrees? The surprising answer is that there is no lower bound: it is possible to rotate the line segment in a set of arbitrarily small area. This question turns out to have important connections to harmonic analysis. We discuss both classical and recent results in this area.

**Udayan B. Darji:** Exploiting Local Entropy Theory

Local entropy theory is a culmination of deep results in dynamics, ergodic theory and
combinatorics. Given a dynamical system with positive entropy, it gives, in some sense, the
location of where the entropy resides. It is a powerful tool that can be applied in a variety of settings.
In this talk, we give an introduction to this theory. We will show how the speaker (with his various co-authors) has been able to apply local entropy theory to settle some problems in continuum theory, and
in dynamics of maps on the space of finite measures. We will also discuss descriptive complexity of some notions of local entropy theory.

**Slawomir Solecki:** Descriptive Set Theory and generic measure preserving transformations

The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$---the topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this was the case.
We will describe in some detail the background touched on above, including connections with Descriptive Set Theory. Further, we will outline a proof of the following theorem that answers the Glasner--Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is

** not ** topologically isomorphic to $L^0$. The proof rests on an analysis of unitary representations of the non-locally compact group $L^0$.

**Libor Vesel:** Extension problems for continuous quasiconvex (or convex) functions

Let X be a normed space and C a convex subset of X. It is well known
that every Lipschitz convex function on C admits a Lipschitz, with the same
constant, convex extension to the whole X. We shall show that when we
substitute convex with quasiconvex:

the same is true if C is a subspace of X;

for a general C with nonempty interior, the extendability depends
drastically on the geometric properties of C.

In the last part, we shall consider problems of extendability of continuous
convex or quasiconvex functions from subspaces.
Let us recall that a function f : C R is called quasiconvex if all its sublevel
sets {x ? C : f (x) ? ?} (? ? R) are convex.

Subjects of the three talks:

1. Non extendability of Lipschitz quasiconvex functions;

2. Uniformly convex sets and extension of uniformly continuous quasiconvex functions;

3. Extension of continuous convex or quasiconvex functions from subspaces.

The talks are based on a joint research with Carlo A. De Bernardi (Universita
Cattolica del Santo Cuore, Milano).