Math 412: Topics in Algebra, Fall 2019
MWF 1:40-2:30pm, Eliot 207Office Hours: M 3:30-4:30pm and Th 2:30-4:00pm in Lib 306
Syllabus | Piazza page | Course notes
E-access to Lam and Szymiczek (requires Reed login).
Final project resources
Please avail yourself of the following resources in preparing your final project:- Assignment details.
- .tex paper template.
- .bib file.
- Advice on how to give a talk from the November 2019 Notices of the AMS.
- Nov 1: Project proposal due via email.
- Nov 22: Draft due.
- Dec 2: Final paper due.
- Dec 2-11: Final presentations in class.
Week 1: September 4 - September 6
- W: Welcome and warmup.
- F: Syllabus and linear algebra review.
Week 2: September 9 - September 13
- M: The quadratic square.
- W: Equivalence, congruence, and isometry.
- F: Regular forms. HW2F due. [Deadline to add classes.]
Week 3: September 16 - September 20
- M: Representation.
- W: Diagonalization I.
- F: Diagonalization II. HW3F due.
Week 4: September 23 - September 27
- M: Hyperbolic spaces.
- W: Witt decomposition and cancellation I.
- F: Witt decomposition and cancellation II. HW4F due.
Week 5: September 30 - October 4
- M: Chain equivalence.
- W: Tensor products of vector spaces.
- F: Tensor products of quadratic forms. HW5F due.
Week 6: October 7 - October 11
- M: Monoids, semi-rings, and group completion.
- W: More group completion.
- F: Viewing of Conway's lecture "$ax^2+bxy+cy^2=n$." HW6F due.
Week 7: October 14 - October 18
- M: The Grothendieck-Witt ring. Take-home exam distributed.
- W: The Witt ring.
- F: The $I$-adic filtration of $GW$ and the "dimterminant" homomorphism; discuss final projects (assignment, paper template, .bib file). Take-home exam due at the start of class. (No homework.)
Fall Break: October 21 - October 25
Week 8: October 28 - November 1
- M: First computations of $GW(k)$ and $W(k)$.
- W: Presentations of $GW(k)$ and $W(k)$.
- F: Nonreal, formally real, and ordered fields. Deadline to submit paper topic proposals via email. HW8F due. (Here are Riley and Yunjia's solutions to 4(a).)
Week 9: November 4 - November 8
- M: Signatures and Pfister's local-global theorem.
- W: Pfister forms.
- F: Multiplicative forms. HW9F due.
Week 10: November 11 - November 15
- M: Function fields and the Hauptsatz.
- W: Quaternion algebras I.
- F: Quaternion algebras II. HW10F due.
Week 11: November 18 - November 22
- M: Quaternion algebras III.
- W: Local fields I.
- F: Local fields II. HW11F and final paper draft due.
Week 12: November 25 - November 29
- M: The Hilbert symbol.
- W: Scharlau and Hilbert reciprocity.
- F: Thanksgiving holiday 🦃.
Week 13: December 2 - December 6
- M: Evan - Small Witt rings. Course evaluations.
- W: Yunjia - Clifford algebras, Henry - Quadratic cryptography.
- F: Meeting in Phys 122. Francis - Prime ideals in the Witt ring, Luke - Milnor $K$-theory.
Week 14: December 9 - December 11
- M: Usman - Naive $\mathbb{A}^1$-homotopy, Riley - 15 & 290 theorems.
- W: Xinling - Hasse principle, David - Trace forms.
Final exam: Wednesday December 18, 9am-12pm in Eliot 207
- Emphasis on material from Weeks 7-12.
- One two-sided sheet of notes allowed; no other resources.
- Turn in final HW at the start of the exam period.
- Reading and finals week office hours: Th 1-2pm, F 10-11am, M 3-4pm, T 3-4pm, and by appointment.
The $\LaTeX$ document preparation system
Poor handwriting? Love escape characters? Too much free time? Try $\LaTeX$!
- $\LaTeX$ at Reed.
- A short guide [pdf] to writing mathematics with $\LaTeX$.
- Change .pdf to .tex on (nearly) any URL accessed from this page to get the $\LaTeX$ source code.
Kyle M. Ormsby