Séminaire de topologie, géométrie et algèbre (archives)

Pendant ma thèse j'ai construit l'espace des modules d'une certaine classe de connexions méromorphes et j'ai étudié le feuilletage isomonodromique induit par la fibration de Riemann-Hilbert. L'espace des modules en question est modelé sur un fibré en droites qui a pour base l'espace des modules des courbes rationnelles irrégulières. Après avoir donné quelques définitions, on étudiera l'espace des modules des courbes, sa compactification à la Deligne-Mumford, et son lien avec les connexions méromorphes en question.

It remains an open question whether nontrivial linear dependences exist among Seifert fibered spheres in the homology cobordism group. In this talk, we consider an infinite family of homology spheres that form the trivial local equivalence class in involutive Heegaard Floer theory, thereby potentially yielding nontrivial dependences between Seifert fibered spheres in the homology cobordism group. We then focus on a survey of filtered instanton Floer theory. Finally, as an application, we prove that these candidates are linearly independent. In particular, we compare the invariants from the two theories: involutive Heegaard Floer theory and filtered instanton Floer theory. If time permits, we also compare them with Pin(2)-equivariant Seiberg-Witten Floer theory. This is joint work with Jaewon Lee (KAIST, Korea).

Le patchwork combinatoire est une puissante méthode utilisée pour construire des hypersurfaces algébriques réelles avec du contrôle sur sa topologie. Dans cet exposé, je discuterai d'une généralisation de cette méthode en codimension supérieure grâce à la notion de structure de phase réelle. En codimension 2, on donne une nouvelle description explicite (basée sur des triangulations, distributions de signes et orientations d'arêtes) des T-variétés (les variétés obtenues par patchwork) proche de la description originale de Viro pour les hypersurfaces. Cette méthode permet d'obtenir une famille de T-courbes maximales dans l'espace projectif de dimension 3. En grande codimension, on présente de nouvelles bornes sur le nombre de composantes connexes qui montrent qu'on ne peut pas obtenir de T-courbes ou T-surfaces maximales

Symplectic fillings of lens spaces were classified by McDuff and Lisca in the early 2000s. A special class of these fillings arise as fillings of lens spaces L(p^2,pq-1), which admit symplectic fillings with vanishing second Betti numbers. In particular, they are symplectic models for rational homology balls B_{p,q}. We study symplectic embeddings of these models into CP2. We show that such embeddings exist if and only if p is a "Markov number" by "elementary" methods. This is joint work with N. Adaloglou, J. Brendel, J. Evans, and F. Schlenk.

Given an isolated singular point in a complex surface, its link is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that reflects the topology of the singularity. The link has a natural contact structure, and one can ask questions about its contact and symplectic topology, for example, try to describe symplectic 4-manifolds that the link can bound. A family of symplectic (even Stein) fillings is provided by possible smoothings of the singularity; it is then interesting to compare algebro-geometric smoothings and general Stein fillings. We will discuss this question for sandwiched singularities, a subclass of rational singularities where smoothings are well understood due to de Jong-van Straten work from 1990s: a dimensional reduction allows to reconstruct all Milnor fibers from deformations of associated singular plane curve germs. We build a symplectic analog of this theory to describe Stein fillings in terms of certain immersed disk arrangements in a very similar way. However, we also show that this method allows to build Stein fillings whose topology is different from that of any Milnor fibers.

The talk is based on joint work with Starkston and Beke-Starkston.

A central question in the topology of distributions is whether there exist maximally non-integrable distributions beyond contact structures for which the h-principle fails, i.e. whether there are genuinely new invariants not dictated by algebraic topology. Most known classes instead satisfy strong flexibility results and are classified by their obvious homotopical data. In this talk I will briefly survey this landscape and then present a recent joint result with Álvaro del Pino and Eduardo Fernández giving a first example of rigidity beyond the contact world: a class of fat (corank- 2) distributions that naturally generalise contact structures, together with their adapted submanifolds, the prelegendrians. Using a canonical contactisation, we construct families of prelegendrians detected by contact-topological invariants, thereby showing that the ℎ h-principle fails in this setting.

In dimensions greater than four, the classification of smooth manifolds is an unsolvable problem, but manifolds can still be classified up to cobordism.

From this perspective, Liouville cobordisms provide a powerful tool for studying contact manifolds in high dimensions. In this talk, I will explain how Liouville cobordisms can be used to construct exact locally conformally symplectic (LCS) manifolds, in particular the LCS mapping tori associated with a contactomorphism. I will then use this construction to study the isomorphism classes of LCS mapping tori and explore their connections with the contact mapping class group.

Quels espaces lenticulaires admettent un plongement lisse et injectif dans le plan projectif complexe ? Combien peut-on en plonger, tels qu'il soient deux à deux disjoints ? La question vient de la géométrie algébrique complexe, où Hacking et Prokhorov donnent une réponse complète en termes de triplets de Markov. Le même résultat est valable en géométrie symplectique, comme montré par Evans et Smith. Topologiquement, la situation est plus riche, et des exemples non-symplectiques ont été donnés par Lisca et Parma. Dans cet exposé, je parlerai de nouvelles obstructions et constructions. Il s'agit d'un travail commun avec Brendan Owens.

Fibered links are those whose complement admits a fibration over the circle. They admit a simple description in terms of an open book, consisting of a fiber surface together with a self-diffeomorphism (called the monodromy), which is the return map of the fibration. This structure allows us to prove several results for fibered links. In this talk I will explain how to extend the concept of monodromy to all links, and show that it can be used to extend previous results to non-fibered links.