Future teaching:
Summer 2026 (Bonn)
Seminar ``Stacks, matrix factorizations, and the D-equivalence conjecture".
We plan to discuss the following papers of D. Halpern-Leistner (one in collaboration with S. Sam): arXiv:1203.0276, arXiv:1601.02030, arXiv:2010.01127
Current teaching:
Winter 2025 (Bonn)
Class ``Derived categories of coherent sheaves" : Tuesday and Friday 8 am-10 am in Zeichensaal, Wegelstrasse 10.
The main reference for the first part of the class (until December) is ``Fourier-Mukai transforms in algebraic geometry" by D. Huybrechts.
In January, we plan to discuss (part of the) the papers ``The McKay correspondence as an equivalence of derived categories" by T. Bridgeland, A. King, M. Reid, and ``Autoduality of compactified Jacobians for curves with plane singularities" by D. Arinkin.
The handwritten notes are very very rough, please use them mainly as indication of what was discussed in class.
Here are typed notes by Tudor Șarpe and Maxim Jean-Louis Brais.
Week 1 (Oct 14 and Oct 17): overview, categories of chain complexes of an abelian category, F-adapted classes, definition of derived category of an abelian category: Notes
Week 2 (Oct 21 and Oct 24): the homotopy category of an abelian category, mapping cones, (bounded below) derived category as homotopy category for categories with enough injectives, the construction of the derived category (Verdier): Notes
Week 3 (Oct 28 and Oct 31): triangulated categories, the derived category is triangulated, derived functors, Serre functors: Notes
Week 4 (Nov 4 and Nov 7): the derived category of coherent sheaves, deriving left-exact (Hom, pushforward) and right-exact (tensor, pullback) functors, adjunction, the projection formula, upper shriek functor and Grothendieck-Verdier duality, the Grothendieck group, the Fourier-Mukai transform: Notes
Week 5 (Nov 11 and 14): spanning classes and generators of a triangulated category, decomposable categories, semiorthogonal decompositions, example of projective line, projective space: Notes
Week 6 (Nov 18 and 21): Beilinson's resolution of the diagonal, brief discussion of Kuznetsov component for a hypersurface, Orlov's blow-up formula: Notes for weeks 6 and 7
Week 7 (Nov 25 and 28): end of blow-up formula, Bondal-Orlov reconstruction theorem.
Week 8 (Dec 2 and Dec 5): end of Bondal-Orlov reconstruction theorem, standard flop (Bondal-Orlov): Notes for flips/flops.
Week 9 (Dec 9 and 12): Mukai flop (Kawamata, Namikawa), criteria for fully faithfulness: Notes for fully faithfulness.
Week 10 (Dec 16 and 19): brief introduction to moduli spaces of sheaves (with focus on the Picard scheme and the moduli of semistable sheaves on a del Pezzo or Calabi-Yau surface): Notes, derived equivalence for moduli of sheaves on K3 surfaces (Mukai) and between and abelian variety and its dual (Mukai).
Week 11 (Dec 23): no class.
Modern Algebra II at Columbia in Spring 2023.
Calculus III at Columbia in Fall 2022.
Calculus II at Columbia in Spring 2022.
Calculus III at Columbia in Fall 2021.
Review of single and multivariable calculus (18.089) at MIT in Summer 2018 (with Kevin Sackel) and Summer 2019.