Koszul duality appears to be a very fundamental phenomenon in algebra, showing up in various different settings and levels of generality, as well as manifesting itself in most areas of mathematics having to do with algebra (algebraic geometry, algebraic topology, representation theory, etc.). In particular, many surprising equivalences of categories could in fact be interpreted as Koszul duality. One of the views on Koszul duality is that it answers a very natural question: when and how an object of an algebraic nature could be recovered from its homological invariants (compare this to the same question asked about a topological space or a manifold)?
In this class I will cover both the theoretical basis of Koszul duality and several prominent applications. For the theory, the main references I will use are the book "Quadratic algebras" by A. Polishchuk and L. Positselski and the survey https://arxiv.org/abs/2207.07063 by the last author. Some applications that I would like to cover are: algebraic bundles on projective spaces (following Bernstein-Gelfand-Gelfand), equivariant cohomology (following Goresky-Kottwitz-MacPherson) and category O of representations (following Beilinson-Ginzburg-Soergel). If time permits, there are many more we can talk about.
Prerequisites for the class is a familiarity with homological algebra and category theory (resolutions, derived functors, abelian categories, derived categories,...). Some background in algebraic geometry, algebraic topology and Lie theory will be helpful for the applications, but I will give an overview of what will be necessary for our purposes.
The class is meeting MWF at 1pm in Gasson 209. On this page I will post my lecture notes and the video recordings are available via the link. All the announcements regarding the class will also be posted on this page. Office hours for the course are available by appointment. Don't hesitate to contact me at ionov@bc.edu if you have any questions or concerns.
8/26, Lecture 1, S-Λ duality, part I, notes.
8/28, Lecture 2, S-Λ duality, part II, notes.
8/30, Lecture 3, S-Λ duality, part III, notes.
9/2, Labor Day, no class.
9/4, Lecture 4, S-Λ duality, part IV, notes.
9/6, Lecture 5, S-Λ duality, part V, notes.
9/9, Lecture 6, Algebraic vector bundles on projective spaces, notes.
9/11, Lecture 7, Bar constructions, notes.
9/13, Lecture 8, Bar constructions finished; Quadratic algebras, notes.
9/16, Lecture 9, Quadratic algebras and cohomology, notes.
9/18, Lecture 10, Minimal resolutions, notes.
9/20, Lecture 11, Koszulness, notes.
9/23, Lecture 12, Hilbert series of Koszul algebras, notes.
9/25, Lecture 13, Koszul complex revisited notes.
9/27, Lecture 14, Koszul complex finisihed; Relative Koszulness I, notes.
9/30, Lecture 15, Relative Koszulness II, notes.
10/2, Lecture 16, Koszul algebras in algebraic geometry, notes.
10/4, Lecture 17, Koszulness and distributive collections of vector spaces; Deformations of Koszul algebras, notes.
10/7, Lecture 18, Koszul algebras with A_0 semisimple, notes.
10/9, Lecture 19, Koszul duality for graded algebras as a derived equivalence, notes.
10/11, Lecture 20, Koszul duality for graded algebras as a derived equivalence finished; Highest weight categories part I, notes.
10/14, Fall break, no classes.
10/16, Lecture 21, Highest weight categories part II; Example I: perverse sheaves, notes.
10/18, Lecture 22, Perverse sheaves finished, notes.
10/21, Lecture 23, Example II: Category O, notes.
10/23, Lecture 24, Category O continued, notes.
10/25, Lecture 25, Parabolic-singular duality, notes.
10/28, Lecture 26, Parabolic-singular duality II, notes.
10/30, Lecture 27, Parabolic-singular duality finished. Mixed Categories, notes.
11/1, no class.
11/4, no class, got sick :-(
11/6, Lecture 28, Mixed Categories II, notes.
11/8, Lecture 29, Mixed Categories III, notes.
11/11, Lecture 30, S-Λ duality in dg-setting, notes.
11/13, Lecture 31, Equivariant cohomology, notes.
11/15, Lecture 32, Equivariant cohomology and Koszul duality, notes.
11/18, Lecture 33, Nonhomogenous quadratic algebras, notes.
11/20, Lecture 34, Nonhomogenous quadratic duality, notes.
11/22, Lecture 35, Nonhomogenous quadratic duality and cohomology, notes.
11/25, Lecture 36, Bar construction for CDGAs; Announcement of Koszul duality for dg-agebras, notes.
12/2, Lecture 37, Twisting cochains, notes.
12/4, Lecture 38, Derived categories of the second kind; Contramodules, notes.
12/6, Lecture 39, Co-contra correspondence; Koszul Triality, notes.
12/9, Lecture 40, Semialgebras and Harish-Chandra pairs, notes.