After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy- Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow. Joint work with F. Cavalletti.

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.

 In a recent paper Friedl, Zentner and Livingston asked when a sum of torus knots is concordant to an alternating knot. After a brief analysis of the problem in its full generality, I will describe some effective obstructions based on Floer type theories.

We develop a structure  theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields  of p-degree one. In turn, we  settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.