To describe the behavior of the iterates of a holomorphic germ f on a complex surface X fixing a (possibly singular) point x0, we are led to study the lifts fπ to birational models Xπ over (X,x0). In general fπ has indeterminacy points: when the fπ-orbits eventually avoid these indeterminacy points, we say that X_π is algebraically stable.
In a joint work with William Gignac, we show the existence of algebraically stable models in this setting (but for one class of exceptions, where no such models exist). The proof relies on fixed point theorems for the dynamics induced on suitable valuation spaces, following Favre and Jonsson.
Les singularités en épissure sont la classe la plus vaste que l'on connaisse de singularités intersections complètes de surfaces complexes dont le bord est une sphère d'homologie entière. Neumann et Wahl conjecturèrent que leurs fibres de Milnor peuvent se construire à partir de fibres de Milnor de singularités plus simples du même type. Je présenterai le contexte dans lequel ils formulèrent cette conjecture et j'expliquerai les étapes de la preuve obtenue en collaboration avec Maria Angelica Cueto et Dmitry Stepanov.
Les nombres de Markov sont des entiers positifs qui apparaissent dans les solutions d'une équation diophantienne, la cubique de Markov
x2 + y2 + z2−3xyz = 0.
Sujet classique de la théorie des nombres, ces nombres sont liés à de nombreux domaines des mathématiques tels que la combinatoire, la géométrie hyperbolique, la théorie de l'approximation et les algèbres amassées.
Dans les années 50, H. Cohn a découvert une relation entre les nombres de Markov et les longueurs de géodésiques simples fermées sur le tore percé. Dans les années 90, avec Igor Rivin, nous avons introduit une méthode qui permet d'estimer le nombre de nombres de Markov inférieurs à $ L> 0 $. L'ingrédient clé en était l'utilisation d'une norme sur la première homologie du tore.
Dans cet exposé nous allons :
- expliquer la géométrie de la norme et comment elle peut être utilisée pour prouver de nouvelles identités pour des longueurs de géodésiques fermées simples
- utiliser la convexité pour donner une nouvelle preuve unifiée de certaines conjectures que Martin Aigner a formulées dans son livre, le théorème de Markov et 100 ans de la conjecture d'unicité.
Given a partial differential equation (PDE), its solutions can be difficult, if not impossible, to compute.
The purpose of the Fundamental theorem of differential tropical (partial) algebraic geometry is to extract from the equations certain properties of the solutions.
More precisely, this theorem proves that the support of the solutions (in $k[[t1, \cdots, tm]]$ with $k$ a field of characteristic zero) of a system of algebraic PDE can be obtained by solving a so-called tropicalized differential system.
Moduli spaces of stable maps in genus bigger than zero include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe. We will look at examples of genus one and two maps of low degree in the projective plane to get a feeling of how complicated these spaces can be.
Afterwords, we will sketch the construction of a modular desingularization of the space of genus 2 maps to projective spaces using combinatorial techniques from tropical geometry and maps from certain exotic curve singularities.
Given a closed 4-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X-int(B^4), with boundary a given knot K in the 3-sphere.
We give several methods to bound the genus of such surfaces in a fixed homology class. Our techniques include adjunction inequalities from Heegaard Floer homology and the Bauer-Furuta invariants, and the 10/8 theorem.
In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disc) in a 4-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
This is joint work with Ciprian Manolescu and Lisa Piccirillo.
In this talk, I will introduce you to two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which come with a handle decomposition compatible with its symplectic structure. I will then show you an algorithm which produces the Weinstein handlebody diagram for the complement of a smoothed toric divisor in a "centered" toric 4-manifold. This is based on joint work with Acu, Capovilla-Searle, Gadbled, Marinković, Murphy, and Starkston.
The goal of the talk is to show you a beautiful matrix and then to explain its geometric significance. This will enable me to explain why two rival geometric interpretations of `Reid's recipe' are equivalent. To begin, I'll set the scene by discussing the classical McKay correspondence in dimension two and I'll go on to discuss how this extends naturally to dimension three. I'll introduce Reid's recipe by studying a resolution of C^3 by an action of the cyclic group of order 19 that gives rise to the beautiful matrix. I'll reveal the geometry that this matrix encodes, and as a result, we'll see that two conjectures for certain toric algebras arising in strong theory are equivalent.
