dans le cadre de la Chaire Poincaré à l'IHP
Institut Henri Poincaré
Fridays Nov. 14, Dec. 12, Jan. 16, and Feb. 13
(2014-15)
This seminar will focus on various aspects of mirror symmetry and related topics in symplectic geometry (Lagrangian submanifolds, Fukaya categories, etc.) and algebraic geometry.
Abstract: Loose Legendrian n-submanifolds, n>1, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are in fact actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian n-spheres in standard contact (2n+1)-space, n > 1, with trivial DGA.
Abstract: A recent preprint of Abouzaid, Auroux, and Katzarkov constructs SYZ mirrors for certain toric Calabi-Yau varieties. In this talk, after reviewing the construction, we examine their construction from the point of view of Homological Mirror Symmetry. More precisely, based upon joint work with Kwokwai Chan and Kazushi Ueda, I'll explain how to recover the coordinate ring of a toric Calabi-Yau variety from the Lagrangian Floer theory of its mirror.
Abstract: Joint work with Yankı Lekili. If we think of CP^{3} as the space of triples of points on the sphere then the Chiang Lagrangian is the subspace of triples with centre of mass at the origin. We will see that it has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5. This calculation involves some general theory, true for all homogeneous Lagrangian submanifolds, and some very specific geometry in CP^{3} involving the twisted cubic.
Abstract: In previous work, we constructed an exotic monotone Lagrangian torus in CP^2 (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it T(1,4,25) because, when following a degeneration of CP^2 to the weighted projective space CP(1,4,25), it degenerates to the central fiber of the moment map for the standard torus action on CP(1,4,25). Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple (i.e. a^2 + b^2 + c^2 = 3abc) there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
Abstract: The SYZ conjecture provides us a geometric way of understanding mirror symmetry. In particular, it gives us a beautiful heuristic picture of a degenerating family of Calabi-Yau manifolds. In 2000, Kontsevich and Soibelman proposed a more "algebraic" approach using the theory of non-archimedean analytic geometry. Since non-archimedean geometry seems "exotic" to many people, I will begin by a brief introduction. I will focus especially on its relations with the differential / symplectic viewpoint. Then I will explain several general results concerning enumerative geometry in this framework. Finally, we will apply the general results to study the enumeration of holomorphic cylinders in log Calabi-Yau surfaces.
Abstract: I will describe a how to lift a Lagrangian from the base of a symplectic fibration to the total space. When the base is a surface, examples include Lefschetz thimbles, matching cycle spheres and matching tori. I will explain how other constructions of Lagrangians in the literature can be viewed from this perspective. These include tori of Auroux, Entov-Polterovich, all tori of Chekanov-Schlenk, and Lagrangians arising from Biran's construction. I will also describe an approach to counting holomorphic discs with boundary on such Lagrangians. This talk is based on joint work with A. Gadbled.
Abstract: I will talk about joint work with Ludmil Katzarkov and Helge Ruddat which proposes a way of thinking about mirrors of varieties which are neither Calabi-Yau or Fano; this includes general type varieties. In particular, we give a mirror construction for a hypersurface X in a toric variety. The mirror Y is of the same dimension as X, but is a singular scheme carrying a perverse sheaf. The cohomology of this perverse sheaf carries a mixed Hodge structure, from which one extracts Hodge numbers which exhibit the usual mirror symmetry with the Hodge numbers of X.
Résumé: La construction de Kenyon et Goncharov associe un système intégrable (une variété de Poisson munie d'une application de Poisson vers l'espace des courbes planes avec un polygone de Newton donné et dont les fibres sont des jacobiennes de ces courbes) à tout polygone convexe aux sommets entiers. On donnera une description explicite de cette construction, de son lien aux groupes de Lie affines, du groupe de symétries discrètes, ainsi que des relations possibles à la géométrie des courbes planes.
Abstract: As a sequel to Mark's talk in the morning, I explain how the naive duality of toric Landau-Ginzburg models that we use for our mirror construction can be fitted into the framework of Fukaya-Oh-Ohta-Ono, Auroux and others. It turns out that our naive potentials receive quantum corrections at higher order. However, these neither affect the critical locus of the potential nor its perverse sheaf or Hodge structure. Hence, the mirror dual we propose via the native potential coincide with the one of the quantum corrected potential. We use the theory of broken lines in the Gross-Siebert program to determine the corrected potential. Some slides concerning a part of this talk can be found at http://www.physik.uni-freiburg.de/~helger/corrected-potentials.pdf
I would like to thank IHP and the Poincaré Chair for providing funding for this seminar.
Denis Auroux
Ecole Normale Supérieure, Salle W
Fridays Sept. 26, Oct. 10, and Oct. 17, 2014
This mini-course (3×3 hours) will aim to give a brief introduction to Fukaya categories and the symplectic geometry of mirror symmetry, especially focusing on homological mirror symmetry and the Strominger-Yau-Zaslow conjecture; the main goal is to give a flavor of the material, and we will omit a lot of technical detail.
The course will meet at Ecole Normale Supérieure (45 rue d'Ulm, DMA, 4th floor, salle W) on Fridays Sept 26 and Oct 10 (11:00-12:30 and 14:00-15:30) and Friday Oct 17 (13:30-15:00 and 15:30-17:00).
The lectures will be either in French or in English depending on the audience.