This seminar will focus on Fourier-Mukai transforms and their applications to classical algebraic geometry and the study of moduli spaces. We will start by introducing derived categories (no prior knowledge will be assumed), then we will define the Fourier-Mukai transform and discuss a few reasons why this is such a powerful tool. After this we will see some applications to the study of Abelian varieties.

This is intended as a friendly learning seminar; we will take the time to introduce the foundations and discuss examples. Everybody is welcome!

**Speakers' meeting:** September 22, 1pm, room MATH-103.

**Schedule:** Wednesday 2-3pm

**First talk:** October 11

**References: **

- Daniel Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press (2006)
- Claudio Bartocci, Ugo Bruzzo and Daniel Hernandez Ruiperez,
*Fourier-Mukai and Nahm transforms in geometry and Mathematical Physics*, Birkäuser (2009) - Alexander Polishchuk,
*Abelian Varieties, Theta Functions and the Fourier Transform*, Cambridge University Press (2003) - Jean-Louis Verdier,
*Des Categories Derivees des Categories Abeliennes*, Asterisque N.239 (1996) - Sergei Gelfand, Yuri Manin,
*Methods of Homological Algebra*, Springer (2003) - Charles Weibel,
*An Introduction to Homological Algebra*, Cambridge Univ. Press (1994) - Richard Thomas, Derived categories for the working mathematician
- Andrei Caldararu, Derived categories of sheaves: a skimming
- Thorsten Holm, Peter Jorgensen, Triangulated categories: definitions, properties, examples

Day | Speaker | Title | References | Room |
---|---|---|---|---|

October 11 | Philip | Introduction to derived categories | Huybrechts | MATH-106 |

October 18 | Stefano | Triangulated categories | Huybrechts | MATH-106 |

October 25 | Alessio | Derived Functors | Huybrechts | MATH-105 |

November 1 | Oliver | The derived category of coherent sheaves | Huybrechts | MATH-106 |

November 8 | Tom | Fourier-Mukai transform: definition and basic examples Orlov's theorem | Huybrechts | MATH-106 |

November 22 | Aeran | Intro to abelian varieties | Polishchuk, Bartocci et al. | MATH-106 |

November 29 | Lucas | Abelian Fourier-Mukai transform | Polishchuk, Bartocci et al. | MATH-106 |

December 13 | ? | Classification of semistable bundles on elliptic curves | Polishchuk, Bartocci et al. | MATH-106 |

- Fourier-Mukai transform in Chow
- Derived categories in physics: B-branes
- Moduli spaces of sheaves on a curve and embedding into the moduli space of its Jacobian