Abstract: Deligne and Illusie proved that if a smooth variety over F_p admits a lift over Z/p² then the truncation of its de Rham complex in degrees < p is quasi-isomorphic to the direct sum of its cohomology sheaves. As a consequence, the Hodge-to-de Rham spectral sequence of a smooth proper liftable variety degenerates, provided that the dimension of the variety is ≤ p.

It turns out that further truncations of the de Rham complex of a liftable variety need not be decomposable. I will describe the obstruction to decomposing the truncation of the de Rham complex in degrees ≤ p in terms of other invariants of the variety, and will use this to give an example of a smooth projective variety over F_p that lifts to Z_p but whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. The proof proceeds by first understanding the obstruction to the formality of the de Rham complex as a commutative algebra (in the appropriate derived sense) using an analog of Steenrod operations from algebraic topology.

Abstract: A tropical curve is a graph embedded in R^2 satisfying a number of conditions. Mikhalkin's celebrated correspondence theorem establishes a correspondence between algebraic curves and tropical curves. This translates the difficult question of counting plane rational curves of degree d through 3d-1 points to the question of counting tropical curves with a given notion of multiplicity through 3d-1 points which can be solved using combinatorics. There are two versions of this correspondence, one for the count of complex curves and one for the real count. In the talk, I will present work in progress joint with Andrés Jaramillo Puentes in which we provide a version Mikhalkin's correspondence theorem over an arbitrary base field k. I will also give a short introduction to counting curves over different fields and tropical geometry.

Abstract: A hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics. The theory in dimension two, namely K3 surfaces, is well understood. The aim of the seminar is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from a point of view of their classification; in particular, about existence of Lagrangian fibrations. We will present some results in dimension four, obtained recently in collaboration with Olivier Debarre, Daniel Huybrechts and Claire Voisin.