Abstract:
Setting the timeline of the events which shaped the Milky Way disc through its 13 billion year old history is one of the major challenges in the theory of galaxy formation. Achieving this goal is possible using late-type stars, which in virtue of their long lifetimes can be regarded as fossil remnants from various epochs of the formation of the Galaxy. There are two main paths to reliably age-date late-type stars: astrometric distances for stars in the turn-off and subgiant region, or oscillation frequencies along the red giant branch. So far, these methods have been applied to large samples of stars in the solar neighbourhood, and in the Kepler field. I review these studies, emphasize how they complement each other, and highlight some of the constraints they provide for Galactic modelling. I conclude with the prospects and synergies that astrometric (Gaia) and asteroseismic space-borne missions reserve to the field of Galactic Archaeology, and advocate that survey selection functions should be kept as simple as possible, relying on basic observables such as colours and magnitudes only.

Abstract:
Asteroseismology has the capability of delivering stellar properties which would otherwise be inaccessible, such as radii, masses and thus ages of stars. When coupling this information with classical determinations of stellar parameters, such as metallicities, effective temperatures and angular diameters, powerful new diagnostics for both stellar and Galactic studies can be obtained. I review how different photometric systems and filters carry important information on classical stellar parameters, the accuracy at which those parameters can be derived, and summarize some of the calibrations available in the literature for late-type stars. Recent efforts in combining classical and asteroseismic parameters are discussed, and the unique- ness of their intertwine is highlighted.

Abstract:
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, quasi-elementary. The main result is that the Picard number of X is at most 11 if X has an elementary rational contraction of fiber type, and 18 if X has a quasi-elementary rational contraction of fiber type.

Abstract:
In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest.

Abstract:
We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has "many" pairs of centrally symmetric vertices.

Abstract:
We announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the 3-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.

Abstract:
Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that are numerically equivalent to a multiple of C. When X is projective, we compare contractible and extremal curves, and we show that every curve in X is numerically equivalent to a linear combination with positive integral coefficients of contractible curves.

Abstract:
Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

Abstract:
Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions.

Abstract:
Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5.