Koszul Duality
V5D1 - Advanced Topics in Topology. Course page on BASIS
Time and Location
Wednesday and Thursday, 12:15 - 14:00
Mathematik Zentrum N 0.003 - Neubau
Endenicher Allee 60, 53115 Bonn
Course description
Koszul duality is a concept from homological algebra that establishes a relationship between augmented associative algebras and augmented co-associative coalgebras, providing a comparison functor between the derived category of an algebra and the derived category of its Koszul dual algebra. Operadic Koszul duality, introduced by Ginzburg and Kapranov, extends this notion to the realm of the theory of operads, which has found many applications in deformation theory, algebraic topology and category theory.
In the course we will begin with the classical algebraic Koszul duality in differential graded setting. Then we will introduce operadic Koszul duality in ∞-categorical languages, beginning with some prerequisites on the theory of ∞-category and higher algebras. We will complement the abstract theories with many applications, such as rational homotopy theory and deformation theory in characteristic 0. If time permits, we will discuss deformation theory in positive characteristic towards the end of the course.
Schedule
Koszul resolutions and Koszul duality
- 09.10 - Introduction and some prerequisites on homological algebras.
- 10.10 - Koszul algebras
- 16.10 - Lecture by Peter Teichner or no lecture
- 17.10 - Lecture by Peter Teichner or no lecture
- 23.10 - Koszul duality of Koszul algebras
- 24.10 - Derived Koszul duality
Operadic Koszul duality due to Ginzburg--Kapranov
- 30.10 - The theory of operads
- 31.10 - Operadic Koszul duality I
- 06.11 - Operadic Koszul duality II
- 07.11 - Operadic Koszul duality III
- 13.11 - 14.11 - No lectures because of the Workshop on Unstable Homotopy Theory
Infinity category and higher algebra
Infinity categorical operadic Koszul duality
Applications
Reference
- Henri Cartan and Samuel Eilenberg. Homological algebra. Paperback ed. Princeton University Press (1999)
- Victor Ginzburg and Mikhail Kapranov. Koszul duality for operads. Duke Math. J. 76.1 (1994), pp. 203–272
- Stewart. B. Priddy. Koszul resolutions and the Steenrod algebra. Bull. Am. Math. Soc. 76 (1969), pp. 834--839 .
- Stewart. B. Priddy. Koszul resolutions. Trans. Am. Math. Soc. 152 (1970), pp. 39--60.
Prerequisites
The course is aimed at Master students with a focus on algebraic topology and algebra. Prerequisites include some basic knowledge of homological algebras and the content of the bachelor courses Topology I and Topology II in Bonn.