Math 290 - Seminar on Mirror Symmetry - Spring 2013

D. Auroux - Tuesdays, 3:30-5pm, Room 736 Evans

Outline

This seminar will revisit the themes of past seminars (see here and here) on geometric aspects of mirror symmetry, the SYZ conjecture, and homological mirror symmetry.

The details will be up to participants; a possible plan is to begin with a review of Lagrangian Floer homology and Fukaya categories, then study Abouzaid and Seidel's work on the wrapped Fukaya category, as well as some papers on mirror symmetry for curves (pairs of pants, genus 2 curves, etc). There will also be a few special talks with guest lecturers.

Prerequisite: some knowledge of symplectic geometry.

What's happening

• 1/18: (Friday, 2pm, in 891) An exotic Lagrangian torus in CP^2 (R. Vianna)
• 1/22: (Tuesday, 4pm in 736) Geometry seminar: String diagrams in Topology, Geometry, and Analysis (K. Wehrheim, MIT)
• 1/29: no meeting
• 2/5: (Tuesday, 3:30pm in 736) Lagrangian Floer homology (D. Auroux) (most of section 1 of ref. 1 below)
• 2/12: Operations in Lagrangian Floer theory (K.Y. Fang) (section 2 of ref.1)
• 2/19, 2/26: Mapping cones, twisted complexes and generators (D. Auroux) (section 3 of ref.1)
• 2/26, 3/5: The wrapped Fukaya category (H. Lee) (ref.2)
• 3/12: Open-closed maps and Abouzaid's generation criterion (Z. Sylvan) (ref.2)
• 3/15: (Friday, 2pm, in 736) Dehn twists and free subgroups of the symplectic mapping class group (Ailsa Keating, MIT)
• 3/19: Example: the cylinder: generation criterion and HMS; Landau-Ginzburg models and derived categories of singularities (D. Auroux)
• 4/2: HMS for the pair of pants (Xin Jin) (ref.3)
• 4/9, 11-12:30 in 891 Evans: The SYZ conjecture (D. Auroux)
• 4/16: 11-12:30 in 891 Evans: Constructing mirrors by SYZ (D. Auroux)
• 4/23: no meeting
• 4/30: (3:30-5pm in 736) Helge Ruddat (Mainz): Toric varieties from a symplectic point of view and SYZ mirror symmetry.
  Abstract: We survey the concepts of symplectic reduction and moment maps in order to explain how a Delzant polytope determines a symplectic manifold with Hamiltonian torus action. Along the work of Guillemin and Abreu, we show how the Legendre transform can be used to furnish this symplectic manifold with a complex structure. We then draw a connection to the Strominger-Yau-Zaslow picture of mirror symmetry and discuss how to go from toric manifolds to degenerate Calabi-Yau manifolds by means of gluing polyhedra along their boundary. Criteria for the smoothability of these degenerate Calabi-Yau manifolds were developed in the Gross Siebert program.
• 5/7: (3:30-5pm in 736) Helge Ruddat (Mainz): Gluing affine manifolds and mirror symmetry of Landau-Ginzburg models by discrete Legendre transform.
  Abstract: Instead of gluing a sphere from polyhedra in order to obtain a degenerate Calabi-Yau manifold as in the last talk, we discuss how one can understand mirror symmetry for when one glues a closed ball, in the simplest case producing the mirror dual of a toric variety which is a Landau-Ginzburg model. In order to study mirror duals of varieties of general type, it is useful to glue a topological space homeomorphic to RⁿxR_≥0. This leads to a duality of Landau-Ginzburg models where the simplest case is the duality of cones in dual real vector spaces. This already gives rich geometric results like the mirror duality of projective hypersurfaces of general type in toric manifolds as in my joint work with Gross and Katzarkov. We explain and state the results.
• 5/9: (Thursday, 3:30-5pm in 736) Helge Ruddat (Mainz): Skeleta and mirror symmetry for varieties of general type.
  Abstract: We give in detail SYZ mirror symmetry for the genus two curve. We show that the Leray numbers of the sheaf of vanishing cycles agree with the Hodge numbers of its mixed Hodge structure. We puncture the mirror dual of the genus two curve until if becomes affine and embed a skeleton in it. Restricting the sheaf of vanishing cycles to this skeleton leads to a mirror dual object of a degenerate genus two curve which can be justified by Leray and Hodge numbers. This gives strong evidence for that there exists an intrinsic T-duality version of mirror symmetry for varieties of general type. If time permits, we also discuss the coherent-constructible correspondence for Calabi-Yau hypersurfaces by means of skeleta.

References

1. D. Auroux, A beginner's introduction to Fukaya categories, arXiv: 1301.7056.
2. M. Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. IHES 112 (2010), 191-240; arXiv:1001.4593.
3. M. Abouzaid, D. Auroux, A. Efimov, L. Katzarkov, D. Orlov, Homological mirror symmetry for punctured spheres, arXiv:1103.4322.
4. P. Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geometry 20 (2011), 727-769; arXiv:0812.1171.
5. D. Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, Surveys in Differential Geometry, Vol. 13, Intl. Press, 2009, 1-47; arXiv:0902.1595.