resume: Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a moduli space of matroids.In this talk, we explain how we can make sense of the moduli space of matroids as an ordered blue scheme. As a first step, we have to review Baker and Bowler's theory of matroids with coefficients in a tract and rephrase this in terms of ordered blueprints. This allows us to employ varieties over F1, the field with one element, to construct the desired moduli spaces, including the universal family of matroids.After explaining the key steps of this construction, we will turn towards consequences for matroid theory. Namely, every matroid defines a point in the moduli space. The corresponding residue field governs the representability of the matroid. More precisely, it represents the realization space of the matroid over a field. This leads to a short and systematic proof of Tutte's celebrated result that every binary and orientable matroid is regular. As far as time allows, we will explain how to (re-)prove other results of similar flavours.

All of this is joint work with Matthew Baker.

13h30-15h00 – Mirko Mauri (Imperial College, Londres)

"The essential skeletons of pairs and the geometric P=W conjecture"

resume: The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties has the homotopy type of a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds.  In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems..