2026/2027 (Autumn) Topics in Mathematical Science
Course themeKoszul algebras
Preliminary lecture plan
| Week | Date | Topic |
|---|---|---|
| 1 | 10-08 | Overview. Reminders on algebras and modules. |
| 2 | 10-15 | Graded rings and modules. Homogeneous ideals. Hilbert series |
| 3 | 10-22 | Tensor algebra. Quadratic algebras |
| 4 | 10-29 | Quadratic dual |
| 5 | 11-05 | Free and projective module. Resolution. |
| 6 | 11-12 | Koszul complex and Koszul algebras |
| 7 | 11-19 | Tor and Ext |
| 8 | 11-26 | Yoneda algebra and Koszul duality |
| 9 | 12-03 | --- NO LECTURE --- |
| 10 | 12-10 | (Catch up lecture) |
| 11 | 12-17 | Hilbert series criteria |
| 12 | 12-24 | Graded modules vs linear complexes |
| 13 | 12-31 | --- NO LECTURE --- |
| 14 | 01-07 | --- NO LECTURE --- |
| 15 | 01-14 | BGG correspondence |
| 16 | 01-21 | TBC |
| 17 | 01-28 | TBC |
| 18 | 02-04 | --- NO LECTURE --- |
Time and Venue Thursday 13:00–14:30, Grad. School of Mathematics Room 409
Evaluation
- There is no examination. Grading is evaluated through homework assignments.
- In each assignment, the 2 highest scoring question will be recorded.
- Final grading is determined by the average of the 5 highest recorded marks.
- Grading scheme is as follows.
- A : 85 ∼ 100%
- B : 70 ∼ 84%
- C : 50 ∼ 69%
- Fail : 0 ∼ 49%
References (In order of preferences)
- A. Polishchuk, L. Positselski, Quadratic Algebras, AMS University Lecture Series 37 (2005).
- A. Beilinson, V. Ginzburg, W. Soergel, "Koszul duality patterns in representation theory," J. Amer. Math. Soc. 9 (1996), 473–527.
- R. Martínez-Villa, "Introduction to Koszul Algebras," Revista de la Unión Matemática Argentina 48 (2007), no. 2, 67–95