Séminaires : CAGe: Combinatorics, Arithmetic and Geometry

Les sujets sont ceux décrits par le titre :). Ils doivent être compris dans un sens large.

Notre objectif est de nous réunir avec une périodicité mensuelle.

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11:30-12:30:  Alex Fink (Queen Mary University of London)

Title: A couple ways matroids are like algebraic varieties

Abstract: Matroids are fundamental combinatorial objects. One way to think about a matroid is as recording the combinatorics of which coordinates can vanish together on a linear subspace. Not all matroids come from linear spaces, but in a surprisingly rich collection of ways they behave like they do: matroids have properties that come from an algebro-geometric construction when the linear space exists, but that still hold when it isn't. Time permitting I'll talk about a few of these, but my target will be new work in progress with Eur and Larson stating some cohomology vanishing theorems that give a second proof of Speyer's f-vector conjecture.

14:45-15:45: Oliver Lorscheid (University of Groningen, Netherlands)

Title: Tits's dream of F_1, combinatorial flag varietes and moduli spaces of matroids

Abstract: The first mathematicians that mentioned the desire for a field F_1 with one element, and of geometry over such an elusive object, was Jacques Tits who pursued this idea in the 1950s. His hope was to explain the analogy between geometries over finite fields F_q and certain incidence geometries that behave like the limit q -> 1. Much later, around 2000, Borovic, Gelfand and White expanded Tits's perspective towards combinatorial flag varieties, which are incidence geometries that stem from matroid theory.

In this talk, we introduce a formalism for algebraic geometry over F_1 that captures all these effects in terms of moduli spaces of flag matroids: finite field geometries emerge as F_q-rational points of these moduli spaces, combinatorial flag varieties arise as rational points with values in the so-called Krasner hyperfield and the Tits's incidence geometries resurface as the subsets of closed points of these moduli spaces. We conclude the talk with an explanation on how algebraic groups generalize to F_1 and in which sense SL(n) acts on the moduli space of flags.

16:00-17:00: Jana Rodriguez Hertz (Sustech, China)

Title: HOW FREQUENT IS THE BUTTERFLY EFFECТ?

Abstract: We pose the following conjecture: The fact that a small amount of disorder (positive Lyapunov exponents) leads to general stable disorder (stable ergodicity and even stable Bernouliness) is the most frequent situation in conservative dynamics. We survey some of the recent advances.