There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
(Joint work with K.Grove and G. Thorbergsson)

Advisor: Natalia Komarova

The classical result of Silver -- construction of the model where the Singular Cardinal Hypothesis fails -- will be presented. The emphasis is on presenting Easton suport iteration and extension of elementary embedding to a generic extension of the universe, which is the key ingredient of the entire construction.

David--Semmes conjecture relates Singular Integrals with Geometric Measure Theory. We are in R^d.
If classical singular integrals (of singularity m) are becoming bounded operators after restriction to an m-dimensional set, does this imply that the set is necessarily ``smooth" (for example, is a subset of m-dimensional Lipschitz manifold)? Everybody believed that the answer is positive. It has been proved for only one case: d=2, m=1. This has been done in the combination of papers by Peter Jones, Pertti Mattila, Mark Melnikov, Joan Verdera, Guy David. However, if d>2 the method explored in these papers did not work, and this was a big roadblock in this part of Harmonic Analysis and Geometric Measure Theory. It still is for d>2, m< d-1. But for any dimension d, and m=d-1, Fedja Nazarov, Xavier Tolsa, and myself, we recently answered positively to this question of Guy David and Steven Semmes.