Title: Betti splitting from a topological point of view

Abstract:
Betti splitting is a Mayer-Vietoris like tecnique to recover the minimal graded free resolution of homogeneous ideals. In this seminar, I will present a topological version of this tecnique, in the context of simplicial complexes. As application, I will show some results establishing that the existence of particular kind of Betti splitting for a triangulation of a closed manifold can express nice topological properties of the underlying space. Several interesting examples will be given.
This is a joint work with Ulderico Fugacci.

Title: The covering minima of lattice polytopes

Abstract: The covering minima of a convex body were introduced by Kannan and Lovasz to give a better bound in the flatness theorem, which states that lattice point free convex bodies cannot have arbitrarily large width. These minima are similar in flavor to Minkowski's successive minima, and on the other hand generalize the covering radius of a convex body. I will speak about recent joint work with Francisco Santos and Matthias Schymura, where we investigate extremal values of these covering minima for lattice polytopes.