Universita di Pisa

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions $n\le 5$. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

The field of transseries was introduced by Ecalle to give a solution to Dulac's problem, a weakening of Hilbert's 16th problem. They form an elementary extension of the real exponential field and have received the attention of model theorists. Another such elementary extension is given by Conway's surreal numbers, and various connections with the transseries have been conjectured, among which the possibility of introducing a Hardy type derivation on the surreal numbers. I will present a complete solution to these conjectures obtained in collaboration with Vincenzo Mantova.
 

We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution).

Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field.

This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma).

In this talk I describe some results obtained in collaboration with

J.F. Lafont and A. Sisto, which concern rigidity theorems for a class of

manifolds which are ``mostly'' non-positively curved, but may not support

any actual non-positively curved metric.

More precisely, we define a class of manifolds which contains

non-positively curved examples.

Building on techniques coming from geometric group theory, we show

that smooth rigidity holds within our class of manifolds

(in fact, they are also topologically rigid - i.e. they satisfy the Borel

conjecture - but this fact won't be discussed in my talk).

We also discuss some results concerning the quasi-isometry type of the

fundamental groups

of mostly non-positively curved manifolds.

I will make a short review of some continous approximations to the Navier-Stokes equations, especially with the aim of introducing alpha models for the Large Eddy Simulation of turbulent flows.

Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations.