Professor A.J. de Jong, Columbia university, Department of Mathematics.

Organizational: The talks will be 2x45 minutes with a short break. Time and place: Fridays 10:30 AM in Room 312.

Introductory lecture topics (more than one topic per lecture):

- Gabriel-Zisman localization.
- A bit on triangulated categories.
- Localizing triangulated categories.
- Homotopy category of complexes in an additive category as a triangulated category.
- Construction of the derived category of an abelian category as in Verdier's thesis.
- Construction of the filtered derived category of an abelian category.
- Bounded below derived category of an abelian category with enough injectives.
- Deligne's general discussion of derived functors in Exposee XVII in SGA4.
- Spaltenstein's paper on unbounded resolutions.

- Neeman's paper on Grothendieck duality.
- Derived category modules over a ring determines the ring.
- Beilinson's decomposition theorem for D^b_{coh}(P^n).
- Bondal-Orlov: D^b_{coh}(X) determines X in some case.
- Orlov's computation of Aut(D^b(A)) with A an abelian variety.
- Orlov's description of the triangulated category associated to a hypersurface singularity in terms of matrix factorisations.
- Bondal-van den Bergh: D^b_{coh}(X) is saturated when X is a smooth proper variety.

- Jean-Louis Verdier,
*Des categories derivees des categories abeliennes,*Asterisque No. 239 (1996) - Pierre Deligne,
*Cohomologie a support propres,*Exposee XVII, SGA4. - Beilinson, Bernstein, Deligne,
*Faisceaux pervers,*Asterisque, 100. - Bondal, van den Bergh,
*Generators and representability of functors in commutative and noncommutative geometry*Mosc. Math. J. 3 (2003), no. 1, 1-36, 258. - Bondal, Orlov,
*Reconstruction of a variety from the derived category and groups of autoequivalences,*Compositio Math. 125 (2001), no. 3, 327-344. - Orlov,
*Derived categories of coherent sheaves on abelian varieties and equivalences between them,*Izv. Math. 66 (2002), no. 3, 569-594. - Orlov,
*Derived categories of coherent sheaves and triangulated categories of singularities,*Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009. - Beilinson,
*The derived category of coherent sheaves on P^n,*Selecta Math. Soviet. 3 (1983/84), no. 3, 233-237. - Thomason,
*The classification of triangulated subcategories,*Compositio Math. 105 (1997), no. 1, 1-27. - Spaltenstein,
*Resolutions of unbounded complexes,*Compositio Math. 65 (1988), no. 2, 121-154. - Neeman,
*Triangulated categories,*Annals of Mathematics Studies, 148. - Neeman,
*The Grothendieck duality theorem via Bousfield's techniques and Brown representability,*J. Amer. Math. Soc. 9 (1996), no. 1, 205-236. - Theo Bühler,
*Exact categories,*Expo. Math. 28 (2010), no. 1, 1-69. - Jeremy Rickard,
*Morita theory for derived categories,*J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. - Amnon Yekutieli,
*Dualizing complexes, Morita equivalence and the derived Picard group of a ring,*J. London Math. Soc. (2) 60 (1999), no. 3, 723-746.