Séminaire de topologie, géométrie et algèbre (archives)

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.

Exotic manifolds are smooth manifolds which are homeomorphic but not diffeomorphic to each other. Constructing exotic manifolds in dimension four has been an active research area in low dimensional and symplectic topology over the last 30 years. In this talk, we will first discuss major open problems and some recent progress in 4-manifolds theory. Then we will discuss our constructions of exotic 4-manifolds via pencils of complex curves of small genus and via symplectic and smooth surgeries. Some of our results that will be presented are joint with A. Akhmedov.

This talk concerns Gabrielov’s rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1970’s. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov’s rank Theorem provides a formal condition for the equality to hold. Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with André Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov’s rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof. I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

Étudier les variétés à travers des dégénérescences est une technique standard en géométrie algébrique. Dans ce projet avec Böhning et von Bothmer, nous appliquons cette technique aux anneaux de Chow. Pour des dégénérescences semistables, nous construisons un anneaux de Chow prelog associé à la fibre spécial qui reçoit une flèche de l'anneaux de Chow de la fibre générique. Comme application, nous démontrons que une décomposition de la diagonale se spécialise en une décomposition prelog de la diagonale. En ce faisant, nous rendons applicables à des familles semistables le critère de Colliot-Thélène et Voisin.

In this talk I will discuss infinite-type surfaces (e.g. surfaces of infinite genus). I will show how infinite-type surfaces can be seen as subsurfaces of themselves in a nontrivial way, how to use this fact to select a class of arcs on a surface and to construct a graph with an interesting action by the group of symmetries of the surface (the so-called mapping class group). Joint work with Tyrone Ghaswala and Alan McLeay.

A consequence of Thom Isotopy Lemma is that the set of solutions of a regular smooth equation is stable under C^1-small perturbations (it remains isotopic to the original one), but what happens if the perturbation is just C^0-small? In this case, the topology of the set of solution may change, but it turns out that the Homology groups cannot "decrease". In this talk I will present such result and some related examples and applications. This theorem is useful in those contexts where the price to pay to approximate something in C^1 is higher than in C^0. For instance in the search for quantitative bounds (here the price can be the degree of an algebraic approximation) or in combination with Eliashberg's and Mishachev's holonomic approximation Theorem (which is C^0 at most).