2025/2026 (Autumn) Topics in Mathematical Science

Course themeQuiver representations and homological algebras
Preliminary lecture plan
Week Date Topic
1 10-02 Overview. Reminders on algebras and modules.
2 10-09 Indecomposablility, Krull-Schmidt, simple, Schur's lemma
3 10-16 Quivers and path algebras and their representations. Duality. Idempotents
4 10-23 Jordan-Hölder, Artin-Wedderburn
5 10-30 Bounded path algebra. Nakayama path algebras vs Topological Data Analysis.
6 11-06 Representations of Kronecker quiver vs linear matrix pencils. Four subspace problem vs Representations of D4-tilde quiver. (TBC: Gabriel's theorem)
7 11-13 Category theory, additive and abelian categories.
8 11-20 Bimodule, tensor Hom adjunction, exactness
9 11-27 --- NO LECTURE ---
10 12-04 Exactness (continued), projective, injective
11 12-11 Morita theory
12 12-18 --- NO LECTURE ---
13 12-25 Resolution, Ext, extension, Tor
14 01-08 --- NO LECTURE ---
15 01-15 Tilting modules, endomorphism algebra
16 01-22 Tilting theory (continued)
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)
  • I. Assem, A. Skowronski, D. Simson: Elements of the Representation Theory of Associative Algebras Vol 1. Cambridge University Press 2006
  • M. Auslander, I. Reiten, S.O. Smalo: Representation Theory of Artin Algebras. Cambridge University Press 1995
  • K. Erdmann and T. Holm: Algebras and representation theory. Springer Undergraduate Mathematics Series, Springer 2018
  • A. Zimmermann: Representation Theory: A Homological Algebra Point of View. Springer 2014