Math 512 - Spring 2004



Week one:
  • Monday (Feb 2): Review of some ring theory (PIDs) 313 problem 5. Review of some group theory 512 problem 4.
  • Wednesday: Gaussian integers form an Euclidean Domain. Non UFD example. Review of group actions, Jordan decomposition. 593 problem 8.

    Week two:
  • Monday (Feb 9): Equivalence relation review, quotient group, quotient ring, simple ring, maximal ideal, Math 512 exam problems 1, 2, Math 593 exam problem 3, Chinese remainder theorem.
  • Wednesday: Problems on SL(2,R), (R=Z, or field F), Math 593 exam problem 7, Bruhat decomposition for SL(2,F), Math 512 exam problem 6, Math 593 exam problem 1.

    Week three:
  • Monday (Feb 16): Symmetry group of Euclidean space. Hyperbolic upper half plane. Action of SL(2,R) on upper half plane. Applications of SL(2,R) generators to prove SL(2,R) preserves the hyperbolic metric.
  • Wednesday: Description of geodesics in the upper half plane with the aid of the SL(2,R) action. From Ash's webtext chapter 4 or Artin's chapter 12. Modules basics: definitions, examples.

    Week four:
  • Monday (Feb 23): Left, right modules, submodules, quotient modules, isomorphism theorems, cyclic modules, direct sums.
  • Wednesday: Finitely generated modules, free modules, rank, universal property of free modules.

    Week five:
  • Monday (Mar 1): Endomorphism rings of modules, matrix rings, opposite rings, submodules of a finite rank free module over a PID.
  • Wednesday: Simultaneous basis theorem. Classification of finitely generate modules over a PID. Application to finitely generated abelian groups.

    Week six:
  • Monday (Mar 8): Uniqueness results, noetherian rings.
  • Wednesday: Noetherian rings, Hilbert basis theorem.

    Week seven:
  • Monday (Mar 15): Applications of Hilbert basis theorem. Group representations. Finite generation of invariant polynomials.
  • Wednesday: Finite generation of invariant polynomials, symmetric polynomials and rational functions.

    Week eight:
  • Monday (Mar 22): Hilbert's Nullstellensatz.
  • Wednesday: Hilbert's Nullstellensatz (continue).

    Week nine:
  • Monday (Mar 29): Varieties in complex 2-space, Zariski topology.
  • Wednesday: Quiz on proof that if R is a UFD, then R[x] is also a UFD.

    Week ten:
  • Monday (Apr 12): Holiday
  • Wednesday: Extension rings, integrality, examples for the integers Z and polynomial rings.

    Week eleven:
  • Monday (Apr 19): Integrality (continue), transcendental degree, Noether normalization theorem, statement and geometric meaning.
  • Wednesday: Noether normalization proof, rings and modules of fractions.

    Week twelve:
  • Monday (Apr 26): Localization.
  • Wednesday: Localization (continue) Nakayama's Lemma.

    Week thirteen:
  • Monday (May 3): Tensor products.
  • Wednesday: Tensor products (continue).

    Week fourteen:
  • Monday (May 10): Tensor products (continue).
  • Wednesday: Tensor products (continue).