Algebraic Geometry, Symplectic Geometry and Physics - weeks 1 and 2

August 20-31, 2007

Centre Interfacultaire Bernoulli

EPFL, Lausanne, Switzerland

**Organizers:** D. Auroux, **L. Katzarkov**, A. Kapustin, A. Okounkov, D. Orlov, D. Salamon

This event is part of the semester-long program on Algebraic, Symplectic Geometry and Physics held at Centre Bernoulli, EPFL, Lausanne, Switzerland. See the main program page for information about registration, accomodation, etc. Reminder: all participants (invited or not) should register. Please fill the registration form.

This event will be followed by a conference on symplectic geometry and physics at FIM, ETH Zurich, September 3-7.

The morning sessions will consist of two 80-minute long talks (minicourses), separated by a coffee break. The afternoons will consist of informal discussions and a few talks (to be scheduled later).

All the talks will take place in **room AAC 006** in **building SG**, near Centre
Bernoulli (on the same level). See the campus map.

Monday August 20 | |

9:30-10:50 | M. Kontsevich (IHES)Motivic Donaldson-Thomas invariants I |

11:30-12:50 | Y. Tschinkel (U. Göttingen and NYU)Rational curves on algebraic varieties I |

14:30 | Y. Soibelman (Kansas State)Walls and wall-crossing |

Tuesday August 21 | |

9:30-10:50 | M. Kontsevich (IHES)Motivic Donaldson-Thomas invariants II |

11:30-12:50 | Y. Tschinkel (U. Göttingen and NYU)Rational curves on algebraic varieties II |

14:30 | A. BondalEnhancements and moduli of complexes on complex analytic manifolds |

Wednesday August 22 | |

9:30-10:50 | D. Auroux (MIT)Lagrangian submanifolds and mirror symmetry I |

11:30-12:50 | D. Orlov (RAS)Derived categories of coherent sheaves |

14:30 | F. Perroni (Univ. Zürich)The cohomological crepant resolution conjecture for P(1,3,4,4) |

Thursday August 23 | |

9:30-10:50 | D. Auroux (MIT)Lagrangian submanifolds and mirror symmetry II |

11:30-12:50 | A. Kresch (Univ. Zürich)Gromov-Witten theory I |

Friday August 24 | |

9:30-10:50 | A. Kresch (Univ. Zürich)Gromov-Witten theory II |

11:30-12:50 | D. Orlov (RAS)Triangulated categories of singularities |

Monday August 27 | |

9:30-10:50 | M. Abouzaid (CMI and MIT)Tropical Geometry and Homological Mirror Symmetry |

11:30-12:50 | S. Shadrin (Univ. Zürich)Tautological relations I: Givental theory |

14:30 | D. Kaledin (RAS)The category of planar trees |

Tuesday August 28 | |

9:30-10:50 | M. Abouzaid (CMI and MIT)Equivalences of chain-level theories |

11:30-12:50 | S. Shadrin (Univ. Zürich)Tautological relations II: BCOV action |

14:30 | A. Kuznetsov (RAS)Categorical resolutions of singularities |

Wednesday August 29 | |

9:30-10:50 | A. Kapustin (Caltech)Holomorphic field theory and electric-magnetic duality I |

11:30-12:50 | A. Kapustin (Caltech)Holomorphic field theory and electric-magnetic duality II |

14:30 | K. Fukaya (Kyoto U.)Mirror symmetry of abelian varieties revisited |

Thursday August 30 | |

9:30-10:50 | G. Mikhalkin (Univ. Toronto)TBA |

11:30-12:50 | G. KerrVanishing cycles of irregular points I |

14:30 | B. Siebert (Univ. Freiburg)Introduction to affine geometry |

Friday August 31 | |

9:30-10:50 | G. Mikhalkin (Univ. Toronto)TBA |

11:30-12:50 | G. KerrVanishing cycles of irregular points II |

14:30 | Y. RuanSymplectic birational geometry |

Mohammed ABOUZAID (CMI, MIT)

Denis AUROUX (MIT) (week 1)

Fedor BOGOMOLOV (CIMS, NYU)

Alexei BONDAL (RAS)

Paul BRESSLER (week 2)

Kenji FUKAYA (Kyoto U.) (week 2)

Kentaro HORI (Toronto) (week 2)

Atanas ILIEV (RAS)

Dmitry KALEDIN (RAS) (week 2)

Ludmil KATZARKOV (U. Miami)

Gabriel KERR (U.Chicago)

Seonja KIM (Chungwoon Univ.) (week 1)

Maksim KONTSEVICH (IHES) (week 1)

Andrew KRESCH (Univ. Zurich) (week 1)

Alexander KUZNETSOV (RAS) (week 2)

François LALONDE (Montreal) (week 2)

Grigory MIKHALKIN (Toronto)

Kaoru ONO (Hokkaido U.) (week 2)

Andrei OKOUNKOV (Princeton) (week 2)

Dmitri ORLOV (RAS)

Andreas OTT (ETHZ)

Fabio PERRONI (Univ. Zurich)

Victor PRZHIYALKOVSKIY (RAS)

Yongbin RUAN (U. Michigan) (week 2)

Dietmar SALAMON (ETHZ)

Sergey SHADRIN (Univ. Zurich)

Bernd SIEBERT (Univ. Freiburg) (week 2)

Yan SOIBELMAN (Kansas State) (week 1)

Yuri TSCHINKEL (CIMS, NYU) (week 1)

Chris WOODWARD (Rutgers) (week 2)

**M. Abouzaid (CMI and MIT)**

Tropical Geometry and Homological Mirror SymmetryI will explain how tropical degenerations can be used to understand some Landau-Ginzburg model. I will focus on examples with a toric flavour.

**Equivalences of chain-level theories**Fukaya and Oh conjectured that the multiplicative structures arising in Morse-Witten theory are equivalent to those arising in classical algebraic topology. I will explain how to prove a version of this statement, as well a similar result relating Morse theory and Floer theory in the cotangent bundle. These results are needed when trying to use tropical methods to prove HMS.

**D. Auroux (MIT)**

Lagrangian submanifolds and mirror symmetryI will discuss some aspects of symplectic geometry relevant to mirror symmetry in the Fano setting. The first talk will focus on the geometry of the moduli space of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kähler manifold. In particular I will show how the Landau-Ginzburg superpotential arises from a count of holomorphic discs, and wall-crossing phenomena lead to so-called quantum corrections. The general features will be illustrated by considering the specific example of the complex projective plane. The second talk will be an introduction to Fukaya categories of Landau-Ginzburg models, and will outline how homological mirror symmetry works on concrete examples such as the complex projective plane and its blowups.

**K. Fukaya (Kyoto U.)**

Mirror symmetry of abelian varieties revisitedPart of this talk is a review of Mirror symmetry of abelian variety with flat Lagrangian submanifold. I will also add several points describing the ideas how to generalize it to the case when Lagrangian submanifold is not flat and how the story is related to the Lagrangian surgery. I will explain how the trancendental approach here is related to the more algebraic approach by Gross-Siebert and Kontsevich-Soibelman.

**A. Kapustin (Caltech)**

Holomorphic field theory and electric-magnetic dualityRecently, it was shown that the main statements of the geometric Langlands program follow from the electric-magnetic duality of N=4 supersymmetric Yang-Mills theory. Certain N=2 Yang-Mills theories are also believed to enjoy electric-magnetic duality. We show that such N=2 Yang-Mills theories can be twisted into holomorphic-topological field theories. Upon compactification on a Riemann surface, duality of N=2 SYM theory implies certain statements about the derived category of coherent sheaves on the moduli space of generalized Higgs bundles. This can be regarded as a generalization of (the classical limit of) the geometric Langlands program.

**G. Kerr**

Vanishing cycles of irregular pointsIn this talk I will describe Lagrangian vanishing cycles which arise from irregular points of a generalized Lefschetz pencil. These points sit on a divisor in the compactification of the original symplectic manifold. Such lagrangian cycles can be put into the framework of a directed Fukaya category and yield invariants of the compactified symplectic manifold. After defining the structures of interest, I will discuss possible applications to mirror symmetry, degenerations of Lefschetz pencils and the symplectic embedding problem.

**M. Kontsevich (IHES)**

Motivic Donaldson-Thomas invariantsRecently together with Y. Soibelman we propose a definition of certain generating series for a very large class of triangulated A-infinity Calabi-Yau categories of dimension 3. It generalizes the classical Donaldson-Thomas invariants counting the virtual number of ideal sheaves of 1-dimensional subschemes in a 3-dimensional Calabi-Yau manifold. Our series depends on two t-structures and takes values in functions on a quantum torus with coefficients in the Grothendieck ring of varieties. For CY3 categories associated with quivers (and superpotentials) one obtains a structure closely related to mutations of clusters, by Fomin and Zelevinsky.

**A. Kresch (Univ. Zürich)**

Gromov-Witten theoryI will give a survey of Gromov-Witten theory, focusing on examples. The first lecture will be completely elementary and will review the basic definitions and some of the early motivating examples in the theory. The second lecture will contain an overview of Gromov-Witten theory of orbifolds.

**B. Siebert (Univ. Freiburg)**

Introduction to affine geometry.This talk is meant as an introduction to affine geometry as needed for applications to mirror symmetry. I am planning to discuss the construction of affine structures from polytopes, the relation to the SYZ-picture and how to construct some singular Lagrangian submanifolds, the concepts of monodromy, positivity and simplicity, the notion of radiance obstruction and its relation to moduli of affine structures, and polarizations.

**S. Shadrin (Univ. Zürich)**

Tautological relations I: Givental theory

Tautological relations II: BCOV actionI am going to discuss different theories of genus expansion of Frobenius manifolds and the universal equations that appear in all known approaches and reflect the topology of the moduli space of curves.

In the first lecture, I'll discuss Givental's theory that associates an analogue of the Gromov-Witten potential to any semisimple Frobenius manifold. We proved with Faber and Zvonkine that any Givental's potential satisfies a huge system of so-called tautological relations that are required in the case we consider the system of correlators of a cohomological field theory. I am going to explain this theorem and discuss its application.

In the second lecture, I'll discuss natural extension of BCOV action that provides a genus expansion for the Barannikov-Kontsevich construction on the B side. Again, we proved with Losev and Shneiberg that this genus expansion satisfies all tautological equations. One of the possible proofs (that I am going to discuss) also involves Givental's theory and a mysterious relation of the quantum master equation to the excessive intersection formula in the moduli space of curves.

**Y. Tschinkel (U. Göttingen and NYU)**

Rational curves on algebraic varietiesI will survey several recent results and constructions concerning rational curves on algebraic varieties, with an emphasis on rationally connected varieties and Calabi-Yau varieties.