Séminaire de topologie, géométrie et algèbre (archives)

In this talk, we show that if the image of an embedded Legendrian in a cosphere bundle under a contact homeomorphism is smooth, then it remains Legendrian and its Maslov class is preserved. Our proof is based on the microlocal sheaf theory, using the continuity of the interleaving distance of sheaves with respect to the C0-distance. This is joint work in preparation with Tomohiro Asano, Christopher Kuo, and Wenyuan Li.

In this talk, I will explain how to relate the Novikov ring and microlocal sheaf theory based on the works of Tamarkin and Vaintrob. This relation has been successfully applied to the irregular Riemann-Hilbert correspondence and symplectic geometry, and recently, Scholze revealed certain potential applications of the relation in analytic geometry. Surprisingly, we notice that the almost ring theory invented by Faltings for p-adic Hodge theory plays a central role in the related construction. After that, I will explain some applications of the idea in symplectic geometry.

This talk is based on a joint work with Tatsuki Kuwagaki.

Des modèles algébriques pour les types d'homotopie rationnels ont été proposés par Quillen en 1969 puis Sullivan en 1977. Le modèle de Quillen, utilisant les algèbres de Lie, permet de décrire les types d'homotopie rationnels simplement connexes, tandis que le modèle de Sullivan, utilisant les algèbres commutatives, décrit les types d’homotopie finis et nilpotents. Plus récemment, Buijs, Félix, Murillo et Tanré d’une part et Robert-Nicoud et Vallette d’autre part, ont étendu le modèle de Quillen au cas non-simplement connexe. Dans cet exposé, on présentera une théorie des invariants de Postnikov pour les algèbres de Lie qui est compatible avec le foncteur d’intégration de Robert-Nicoud et Vallette. On en déduira une intégration de la cohomologie de Chevalley-Eilenberg en la cohomologie d’espaces à coefficients locaux. Si le temps le permet, on exposera la motivation initiale pour construire une telle théorie des invariants de Postnikov, à savoir, une conjecture proposée par Félix et Tanré, qui relie les modèles en homotopie rationnelle et le modèle de Loday pour les n-types d’homotopie, i.e. les Cat^n-groupes. Si le temps le permet également, nous tenterons d’expliquer en quoi une théorie des invariants de Postnikov peut permettre d’axiomatiser les foncteurs d’intégrations.

Open book decompositions—known in various contexts as global Poincaré–Birkhoff sections, relative mapping tori, Milnor fibrations, fibered links, and spinnable structures—have arisen independently across several areas of mathematics. Introduced by Thurston and Winkelnkemper, they became a central tool in 3-dimensional contact topology through the groundbreaking work of Giroux, who established a one-to-one correspondence between contact structures up to isotopy and open book decompositions up to positive stabilization.

The strength of this correspondence lies in its combinatorial nature: an open book is determined by a mapping class group element of a surface with boundary, which in turn can be expressed in terms of Dehn twists along simple closed curves. As a result, problems in contact topology can be translated into combinatorial questions about curves on surfaces. This perspective enables explicit computations and offers a powerful framework for proving structural results.

In this talk, I will sketch a proof of the Giroux correspondence using the interplay between open book decompositions and Heegaard splittings. This is joint work with Joan Licata.

A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring, to other algebraic structures (encoded by operads, possibly colored, or by properads), and to address a more general problem: the existence of homotopy equivalences between algebraic structures. I will prove new formality and homotopy equivalence results based on this obstruction theory, presenting applications in several domains such as algebraic geometry, representation theory and mathematical physics.

It is nowadays trendy to count not any more with positive integers (as complex people do) nor with integers (as real people do), but with quadratic forms. This talk will address the problem of enumerating rational curves in algebraic surfaces from this quadratic point of view. I will give a conjectural expression of these geometric quadratic invariants only in terms of Gromov-Witten and Welschinger invariants. In other words quadratic invariants over any field should be determined by the two special fields C and R. This is a joint work with Johannes Rau and Kirsten Wickelgren.

In 1979, Litherland proved that torus knots are linearly independent in the concordance group. We study a higher-dimensional version of this problem, by considering links of Brieskorn-Pham singularities. This is joint work in progress with our very own Oğuz Şavk.

We prove the quilted Floer cochain complexes form A infinity n-modules over the Fukaya category of Lagrangian correspondences. Then we prove that when we restrict the input to mapping cones of product Lagrangians and graphs, the resulting bar-type complex can be identified with bar complex from ordinary Floer theory. As an application we use a family version of quilt unfolding argument to prove two long exact sequences conjectured by Seidel that relates the Lagrangian Floer cohomology of a collection of (possibly intersecting) Lagrangian spheres and the fixed point Floer cohomology of composition of Dehn twists along them.

It is known that a manifold diffeomorphic to a K3 surface admits an almost toric fibration (ATF). However, given a specific symplectic form on a K3 it is unclear if it admits an almost toric fibration with lagrangian fibres for the given form. In this talk, we prove that when a Kähler K3 surface admits a Type III Kulikov degeneration with a symplectic form taming the complex structure, the symplectic form admits an ATF whose base is the intersection complex of the degenerate fibre. Furthermore, we shall show that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric Kähler form, admits such a symplectic Kulikov model.

This is based on joint work with Yoel Groman.

Two smoothly embedded surfaces in a smooth 4-manifold are called exotic if they are topologically isotopic but not smoothly isotopic. The phenomenon of exotic surfaces in dimension 4 is an interesting topic in low-dimensional topology. In this talk, we start with an introduction to equivariant knot concordance theory in the context of exotic disks. Then we recall how involutive Heegaard Floer theory can work as a machinery for equivariant concordance. Then we demonstrate our recent progress on equivariant concordance of Whitehead doubles, which produces exotic disk pairs.

This is joint work with Sungkyung Kang and JungHwan Park.