MATH 221: Algebra

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Location: Science Center 310 
Class Meeting Time: M., W., F., at noon
Instructor: Hiro Lee Tanaka

Hiro's Office: Science Center 341 (in the back of the Birkhoff Math Library) 
Office Hours: Tuesdays, 1 PM - 2:30 PM and 4 PM - 5:30 PM.
Morgan's Office Hours: Tuesdays, 6-7 PM, Sci Center 411.

Here is the syllabus: 2017-fall-221-syllabus.pdf


This is a graduate-level algebra course. We'll start with the representation theory of finite groups, then do some basic ring theory, and then do representations of Lie groups. We will in particular cover the topics required of the Harvard algebra qualifying exam for graduate students, which can be found here.

Announcements

Mon, Sep 11. See Announcement about assumptions on the homework problems.

Mon, Sep 11. Morgan's Office Hours for this week: Tomorrow, Sept 12 (Tuesday), 6-7 PM, Room 411.

Sat, Sep 2. There was a blatant mistake on my part on the universal property of tensor product--as pointed out in class, i should be bilinear. I've updated the notes with red text to reflect this, with brief explanations as to why this is necessary.

Fri, Sep 1. Homework one will be due in two Wednesdays, as we won't have covered enough of the material. (An auspicious start!) Homework two will be due that same Wednesday, but with a little less material than I planned earlier.

 Lectures, Notes, and References

Here is a running list of past and upcoming lecture topics. "Fulton and Harris" refers to their book on representation theory, and "Serre" refers to the book "Linear representations of finite groups" by Serre. 

Click here to ask in-class questions, or questions in general.

  1. Wed, Aug 30. First Class. G-representations. Tensor products. Duals. Universal properties, coproducts. Resource (O'Neil). Resource (Conrad). Notes. 
  2. Fri, Sep 1. More on tensor products (see Conrad notes above). Duals and group representations. [Not covered, to be covered later: Schur's Lemma. Complements to submodules and irreducible representations. Abelian groups and one-dimensional representations.] Fulton and Harris Chapter 1 or Serre 2.2. Notes.
  3. Wed, Sep 6. Schur's Lemma. Complements to submodules. Irreducible reprentations. Abelian groups and one-dimensional representations.  Fulton and Harris Chapter 1 or Serre 2.2. Notes.
  4. Fri, Sep 8. Hiro is out of town; exercise session. Fulton and Harris Chapter 1 or Serre 2.2 and Serre 5. Here is the handout. If you have some in-class questions, ask! How was the exercise session? Fill out this survey!
  5. Mon, Sep 11. Inner product on space of characters. Orthonormality of irreducible representations. Fulton and Harris Chapter 1, Serre 2.2. In-class questions. Here are the notes.
  6. Wed, Sep 13. More on orthonormality as necessary. Fulton and Harris Chapter 1, Serre 2. Here are the notes.
  7. Fri, Sep 15. Some wrap-up of finite group representations. Finish proof of main orthonormality theorem. Here are the notes. 
  8. Mon, Sep 18. Character tables and examples. Alt^2 and Sym^2.  Fulton and Harris Chapter 1, Serre 1 and 2. Here are the notes.
  9. Wed, Sep 20. Begin some commutative algebra. Noetherian property. Motivation from algebraic geometry. Ideals as subspaces. Zorn's Lemma and Noetherian rings. Hilbert basis theorem. Atiyah-MacDonald Chapters 6,7. Here are the notes. Some historical context for the Hilbert basis theorem from Kendig's book.
  10. Fri, Sep 22. More commutative algebra. Modules over Noetherian rings, localization, tensor product. Atiyah-MacDonald Chapters 2 and 3. Here are the notes.
  11. Mon, Sep 25. Prime ideals, maximal ideals as points. Atiyah-MacDonald Chapter 1 and its exercises. Notes.
  12. Wed, Sep 27. More on the Zariski topologies. Tensor products as intersections. Atiyah-MacDonald Chapter 1 and its exercises. Notes.
  13. Fri, Sep 29. Tensor products as intersections. Some basics of category theory, examples. Notes.
  14. Mon, Oct 2. More category theory, more examples. Notes.
  15. Wed, Oct 4. More category theory, more examples. Notes.
  16. Fri, Oct 6. Monoidal categories. Notes.
  17. Wed, Oct 11. Limits and colimits. Notes.
  18. Fri, Oct 13. Toward the Nullstellensatz. Notes.
  19. Mon, Oct 16. Actually proving the weak Nullstellensatz. Zorn's Lemma. Notes.
  20. Wed, Oct 18. Actually proving the Nullstellensatz. Radicals. Jacobson rings.  (Atiyah-MacDonald Chapter 7, Exercise 14. Chatper 5, Exercise 22-23.) Notes.
  21. Fri, Oct 20. Proving the Nullstellensatz. Jacobson Rings. (Atiyah-MacDonald Chapter 7, Exercise 14. Chatper 5, Exercise 22-23.) Notes.
  22. Mon, Oct 23. R Jacobson => S Jacobson if S is a fin-gen R-alg. Notes.
  23. Wed, Oct 25. Krull Dimension. Artin rings. (Atiyah-MacDonald Chapter 8.) Notes.
  24. Fri, Oct 27. More on Artin rings. Notes.
  25. Mon, Oct 30. Integral dependence, closure. Toward curves. (Atiyah-MacDonald Chapter 5.) Notes.
  26. Wed, Nov 1. Curves again. Notes.
  27. Fri, Nov 3. Gauss lemma, Eisenstein criterion. (Lang, Chapters 7 and 8.) Notes.
  28. Mon, Nov 6. Basic field facts. Extensions, degrees, algebraicity, algebraic closures, separability. (Lang, Chapters 7 and 8.) Notes.
  29. Wed, Nov 8. More basic field stuff. (Lang, Chapters 7 and 8.) Separability. Splitting fields. [Not covered: Splitting fields.] Notes.
  30. Fri, Nov 10. Extension lemma for splitting fields. [Not covered: Fundamental theorem of Galois Theory.] Notes.
  31. Mon, Nov 13. Galois correspondence and Galois extensions. Notes. 
  32. Wed, Nov 15. Primitive Element Theorem. Notes. 
  33. Fri, Nov 17. Fundamental Theorem of Galois Theory. Notes.
  34. Mon, Nov 20. Galois theory and geometry of curves. Notes.
  35. Mon, Nov 27. Lie groups and Lie algebras, I. Examples and Definitions of Lie algebras. [Not covered: Geometry of Lie groups, exponential maps, reducing everything to Lie algebra representations.] Notes.
  36. Wed, Nov 29. Lie groups and Lie algebras, II. Example: Irreducible representations of SL(2,C), SU(2), and SL(3,C).
  37. Fri, Dec 1. Last day of classes. (Lie groups and Lie algebras, III?)

 

Course Summary:

Date Details Due