Working seminar on Fourier-Mukai transforms

Fall 2017

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

Tentative Schedule: also availabe on Google calendar

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 30	Lucas	Abelian Fourier-Mukai transform	Polishchuk, Bartocci et al.	MATH-106
December 20	?	Classification of semistable bundles on elliptic curves	Polishchuk, Bartocci et al.	MATH-106

Additional topics for next semester:

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