# Sala Seminari (Dip. Matematica)

## Continuous family of conserved energies for NLS, KdV and mKdV

In this talk based on joint work with Kienzler and Vazquez I will

explain how Caffarelli's strategy of proving regularity for free

boundary problems can be implemented for the porous medium equation.

Elements of the proof is a global existence result for Lipschitz

perturbations of flat fronts of C. Kienzler following my strategy with

Tataru for the Navier-Stokes equations, a careful construction of

comparison solutions, and Gaussian estimates in a subelliptic setting

## Torsion in negative curvature

A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We extend this theorem to the torsion part of the homology in all dimensions d > 3. From Gromov’s work it is known that in dimension 3 the size of torsion homology cannot be bounded in terms of the volume. In dimension 4 we give a somewhat precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d > 4 up to homeomorphism.

## Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

## How to "synthesize" sub-fixed point sets into the global one just by looking inside groups?

Consider a group G action on a metric space X by isometries, and subgroups M_1,..., M_l. One interesting and important problem is to study conditions under which non-emptyness of each M_i-fixed points ensures that of global (G-)fixed points. One extreme case is where G is a simple Lie group, X is an origin-excluded Hilbert space, and the action is given by a (strongly continuous) unitary representation. Then, the Howe--Moore property, based on the Mautner phenomenon, implies that for each(!) non-compact closed subgroup M, the existence of M-fixed points suffices that of G-fixed points.

## Seminario di Algebra e Geometria - "Spectral sequences in local cohomology"

Abstract. Spectral sequences are often applied to compute local cohomology functors.

In this talk I’m going to review their use in order to calculate local

cohomology from the primary decomposition of an ideal I in a commutative Noetherian ring R. By one hand, we shall deal with the

computation of several generalized local cohomology functors supported on I. On the other hand, we will be mainly concerned with

the computation of the local cohomology of R/I.

## A direct approach to an epiperimetric inequality for Free-Boundary problems

## Up to a small error, all functions are s-harmonic

## Nonlocal phase transitions and minimal surfaces

## Dynamics of the critical nonlinear wave equation.

Il professor Thomas Duykaerts (Parigi 13) terrà un minicorso di tre lezioni dal titolo Dynamics of the critical nonlinear wave equation.

Programma: 16 Dicembre, 16-18 Prima lezione

19 Dicembre, 10-12 Seconda lezione

20 Dicembre, 10-12 Terza lezione

Luogo: Aula Seminari