Math 249C Homepage

Math 249C: Perverse sheaves (Spring 2020)

Please see Canvas for the lecture room and the recording. E-mail me if you cannot access it (non-Stanford-affiliate also welcomed).

Instructor: Cheng-Chiang Tsai
Lectures: MWF 9:30am - 10:20am.

Topic: Perverse sheaves as a topological tool, with minor focus on applications towards representation theory and number theory. The audience may think of it as an upgrade of classical sheaf cohomology to better deal with singular algebraic varieties. The course will have an emphasize on actually getting ourselves familiar with the tool, like compute this and that perverse sheaves, resembling the time when we computed this and that (co)homology groups in our childhood.
We will have out setting on algebraic varieties over C, but occasionally wave hands about l-adic cohomology setting where most applications Cheng-Chiang knows live; people familiar with l-adic cohomology should have little problem moving the result there.

Prerequisites: We assume basic knowledge about algebraic varieties over $\mathbb{C}$, and about singular and sheaf cohomology. Some familiarity with derived category will be desirable.

Textbook: Sheaves on Manifold, by Masaki Kashiwara and Pierre Shapira. It is available here with Stanford ID.
Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform, by Reinhardt Kiehl and Rainer Weissauer. It is available here with Stanford ID.

List of lectures:

DateTopicReference and remarkLink to the slideVideo
4/6Overview for the first half of the quarterUpdated 4/6, 11:15am.0406.pdf
4/8Review on derived categoryUpdated 4/8, 11:15am.0408.pdf0408.mp4
4/10(Derived) push-forwardUpdated 4/10, 11am.0410.pdf0410.mp4
4/13Problem Session 1See the proposed problem set here with a solution to P6.
4/15Propr push-forward and proper base changeCheng-Chiang was confused about the direction of the natural morphism regarding base change. But the slide was just correct.0415.pdf0415.mp4
4/17Verdier duality, IUpdated 4/17, 5pm.0417.pdf0417.mp4
4/20Verdier duality, II0420.pdf0420.mp4
4/22Verdier duality, IIILast updated 4/22 evening.0422.pdf0422.mp4
4/24Smooth morphism and Mayer-Vietoris0424.pdf0424.mp4
4/27Very sorry - meeting has to be cancelled due to Cheng-Chiang's illness.
4/29Cohomology on [0,1], retract, and dualityUpdated 4/30, 6pm to fix the gap in homotopy.0429.pdfSome technical issue to be resolved ...
5/1Duality and constructibilityWarning: Essential mistake for the important proof on last slide.0501.pdf0501.mp4
5/4Problem Session 2Problem set is here. A problem added 4/29, 10:45am.0504.mp4
5/6Biduality and perverse t-structure0506.pdf0506.mp4
5/8Perverse t-structure is a t-structureHuge amount of small typos fixed 5/8, 10:40am0508.pdf0508.mp4
5/11Perverse extensions and simple objectsExpanded 5/20, 9am.0511.pdf0511.mp4
5/13See last.0513.mp4
5/15Deligne's formula for middle extensionSee last.0515.mp4
5/18Problem Session 3Problem set is here0518.mp4
5/20Miscellaneous about perverse sheavesSee 5/11.0520.mp4
5/22(Semi-)Small maps and the Decomposition TheoremMistakes in slide corrected 5/22, 1:20pm.0522.pdf0522.mp4
5/25Problem Session 4Problem set is here
5/27, 29Decomposition theorem and l-adic sheavesUpdated 5/29, 9am.0527.pdf0527.mp4, 0529.mp4
6/1, 6/3, 6/5, 6/8Springer theory and characters of G(F_q)Updated 6/8, 9am.0601.pdf0601.mp4, 0603.mp4, 0605.mp4, 0608.mp4
6/10Miscellaneous topics0610.pdf0610.mp4

The slides are subject to random updates. If you see it remarked to be updated after the lecture, typically that is the last update.