The Thurston norm of a 3-manifold measures the minimal
complexity of surfaces representing homology classes. We will show how for some classes of 3-manifolds the Thurston norm can be determined easily using Fox calculus.
This is joint work with Stephan Tillmann.

In questa conferenza verranno esaminati alcuni dei contributi di Federigo Enriques
alle varie discipline di cui si e' occupato (matematica, filosofia, storia delle scienze,
didattica, ecc.) sia come esito di una serie di ricerche maturate all'interno della
cultura italiana e, piu' in generale, europea nella seconda meta' del XIX secolo, sia
come punto di partenza per sviluppi successivi fino ai giorni nostri.

Abstract:
In 1981, Hitoshi Nakada introduced a family of continued fraction maps, and studied
their natural extensions. These are the Nakada alpha-expansions, which are defined for a
parameter alpha between 0 and 1. These alpha-expansions played a key role in the revival of
the interest in continued fraction expansions, and are up to today subject of thorough investigations.
Using some extremely basic ideas called ‘insertions’ and ‘singularizations’ we will show
that there is a strong relation between alpha-expansions for various values of alpha-expansions.
In this talk I will show how far these ideas can be carried over, and how they can be used in
other settings, e.g. for the so-called ‘Rosen fractions.’

The moduli space M_g of Riemann surfaces of genus g is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its
Dolbeault cohomological dimension (i.e. the highest k such that
H^{0,k}(M_g,E) does not vanish for some holomorphic vector bundle E on M_g). The conjecturally optimal bound is g-2, which is verified for g=2,3,4,5.

In this talk, I will show that such dimension is at most 2g-2.
The key point is to show that the Dolbeault cohomological dimension of each stratum of the Hodge bundle is at most g (still non-optimal bound).
In order to do that, I produce an exhaustion function, whose complex Hessian has controlled index: in the construction of such a function basic geometric properties of translation surfaces come into play.

Richiamiamo la costruzione dei luoghi base asintotici
(luogo base stabile, luogo base aumentato e luogo base ristretto)
per fibrati in rette,
e definiamo in maniera analoga dei luoghi base asintotici per fibrati
vettoriali,
collegandoli ad alcune proprieta' di positivita' esistenti nella letteratura
(positivita' debole di Viehweg, ampiezza quasi ovunque, altre condizioni).
Mostriamo una costruzione alla Iitaka per le mappe di valutazione per
fibrati vettoriali.
Si tratta di un lavoro in corso, effettuato in parte in collaborazione
con S. Urbinati, in parte con
T. Bauer, S. Kovács, A. Küronya, e T. Szemberg.