Amoebas A(X) are images of algebraic varieties X in logarithmic coordinates and were introduced by Gelfand, Kapranov, Zelevinsky in their study of discriminants. From a "tropical" point of view, they appear as intermediate objects during the process of passing from the classical algebraic geometry to the piece-wise linear, combinatorial world of tropical geometry. However, basic properties of amoebas, even their dimensions, are not well-understood. In my talk, I will review some results and present a new formula computing dim(A(X)), settling a conjecture by Nisse and Sottile. As a corollary, this formula implies that the amoeba dimension only depends on the tropicalization/Bergman fan of X. This is joint work with Jan Draisma et Chi Ho Yuen.

First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen's and A.Latosiński's) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green's operator on RIemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

On considère un produit tordu (skew-product) d'une application dilatante du cercle par une fonction tau. C'est un modèle simple de dynamique partiellement hyperbolique, faisant intervenir une direction neutre comme le font les flots. Les résonances de Ruelle sont les valeurs propres de l'opérateur de transfert dans des espaces appropriés, elles fournissent des informations sur la dynamique (décroissance des corrélations). Dans ce modèle l'opérateur de transfert se réduit par analyse de Fourier en une famille d'opérateurs, dont nous étudions la distribution des traces (plates) des itérés dans une limite semi-classique, lorsque la fonction tau est aléatoire.

The horofunction boundary was introduced by Gromov in the late 1970s as a general way of compactifying metric spaces. I will describe this boundary in the case of Teichmüller space, and discuss what it tells us about the geometry of this space.

Classical knot theory is the study of smooth 1-manifolds in R^3. Starting from a beautiful construction by Rudolph that relates notions of positivity for knots with the study of algebraic curves in C^2, we discuss applications of knot theory to complex curve questions and vice versa.

Une célèbre conjecture de Kontsevich relie la catégorie des faisceaux cohérents sur une variété algébrique à la catégorie de Fukaya d'une variété de symplectique : il s'agit du phénomène de symétrie miroir. Je présenterai une nouvelle construction de la catégorie de Fukaya enroulée d'un domaine de Liouville ainsi que quelques avantages de cette construction.