Math 412, Fall 2019

Math 412: Topics in Algebra, Fall 2019

MWF 1:40-2:30pm, Eliot 207

Office 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.

Project deadlines are as follows:

Nov 1: Project proposal due via email.
Nov 22: Draft due.
Dec 2: Final paper due.
Dec 2-11: Final presentations in class.

Also plan to give a practice presentation at a time scheduled with the instructor approximately one week before your final presentation.

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.