This course intends to give a concise introduction into the concepts of modern homological algebra and to highlight their applications. To this end, two hours of lectures will be followed by a one hour discussion of examples and applications.

Definitions, structure, and representation theorems for Abelian Categories. Resolutions by projectives, injectives and cohomology theories. Hochschild Cohomology, Gerstenhaber Algebras, Super Poisson and Super Lie algebras. Triangulated Categories, exact functors Fourier-Mukai transforms and other equivalences. Stable theories: Tate cohomolgy, periodicity. The homological mirror conjectures.

Prerequisite: Core course in Algebra required, interest in algebraic geometry helpful.

References:
Bass, Hyman . Algebraic $K$-theory.(W. A. Benjamin, Inc., New York-Amsterdam 1968 xx+762 pp.)
Bourbaki, Nicolas . Algebre. Chapitre 10. Algebre homologique. (Masson, Paris, 1980. vii+216 pp. ISBN: 2-225-65516-2)
Cartan, Henri; Eilenberg, Samuel . Homological algebra.(With an appendix by David A. Buchsbaum.Reprint of the 1956 original.
Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. ISBN: 0-691-04991-2)
Gelfand, Sergei I.; Manin, Yuri I. Methods of homological algebra. 2nd ed. Springer Monographs in Mathematics. (Springer-Verlag, Berlin, 2003. xx+372 pp. ISBN: 3-540-43583-2)
Hilton, P. J. ; Stammbach, U. A course in homological algebra. 2nd ed. (Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. ISBN: 0-387-94823-6)
Mac Lane, Saunders. Homology. Reprint of the 1975 edition.Classics in Mathematics. (Springer-Verlag, Berlin, 1995. x+422 pp. ISBN: 3-540-58662-8)
Mac Lane, Saunders . Categories for the working mathematician. 2nd ed. Graduate Texts in Mathematics, 5. (Springer-Verlag, New York, 1998. xii+314 pp. ISBN: 0-387-98403-8)
Rotman, Joseph J. An introduction to homological algebra.Pure and Applied Mathematics, 85. (Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. xi+376 pp. ISBN: 0-12-599250-5)
Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. (Cambridge University
Press, Cambridge, 1994. xiv+450 pp. ISBN: 0-521-43500-5; 0-521-55987-1)