Number Theory MATH 354

Class information

Number Theory MATH 354

Spring 2017

Instructor: Michael Magee

(firstname).(lastname)@yale.edu

Office: Dunham Labs 418

Office hours: TBA

Textbook

A Friendly Introduction to Number Theory, 4th Edition, Joseph H. Silverman.

A Classical Introduction to Modern Number Theory, Kenneth Ireland and Michael Rosen

Final Project

This is going to be a central part of the class and a valuable chance for you to learn how to write and present some mathematics. By around halfway through the class, you will have decided, in consultation with me, on a topic in Number Theory that you will study in detail for your final project.
The expectation for this project is that you give a clear exposition of the chosen topic, written with LaTex, that is around 10-15 pages long. You will also give a presentation to the class on your topic.
In order to get your writing up to a high standard, you will submit several drafts of your project to me at scheduled dates and I will give you feedback for your next draft. By the end, you should have a nice piece of mathematical writing.
I can also help coach you on your presentation skills.

Here are some suggested topics, but you are also free to come up with your own.
They are loosely categorized, and skewed according to my own tastes.
Algebraic
The group law on an elliptic curve.
The class group of a number field
Analytic
A history of prime number theorems.
The circle method and Vinogradov’s theorem
Diophantine approximation
The Lagrange spectrum and Markoff numbers
Computer science related
Primes are in P and related topics.
Elliptic curve cryptography
Miscellaneous
The geometry of numbers
Random model for the Mobius Function
Cohen-Leinstra Heuristics
The Sato-Tate conjecture
GUE and the spacings of Riemann zeros (related to Physics)
Special values of Riemann zeta (related to Physics)

Grading

40% Homework

40% Final Project

20% Final Presentation

Homework/Project assignments

HW 1.
From Ireland and Rosen Chapter 1:
Questions 1.3, 1.10, 1.13,
1.16, 1.18, 1.25, 1.30

HW 2.
From Ireland and Rosen Chapter 2:
Questions 2.1, 2.2, 2.3, 2.6,
2.10, 2.13, 2.25, 2.26

HW 3.
From Ireland and Rosen Chapter 3:
Qs 3.1, 3.4, 3.5, 3.8,
3.9 (typo here, use 3.8 not 3.7),
3.15, 3.16, 3.22, 3.23

HW 4.
From Ireland and Rosen Chapter 4:
Qs 4.1, 4.3, 4.6, 4.12,
4.15, 4.20, Prove Proposition 4.2.4

HW 5.
From Ireland and Rosen Chapter 5:
Questions 5.2, 5.12, 5.14, 5.23, 5.24, 5.25, 5.26, 5.27, 5.28
Begin work on
Project Assignment 1:
Write a 3 page paper on your chosen topic.
This will form a starting point for your final paper.
Things to aim for:
A title.
A introduction with a description of the topic,
and maybe a little history.
A clear statement of a main theorem in the area.
A proof of something, maybe a preliminary result
or a result that you use in the introduction.
References.

HW 6

Schedule

We meet for Lecture in LOM 205, on Tuesday and Thursday from 2.30pm to 3.45pm.

Lecture 01. Jan 17. (Tu)

Lecture 02. Jan 17. (Th)

Lecture 03. Jan 24. (Tu)

Lecture 04. Jan 26. (Th)

Lecture 05. Jan 31. (Tu)

Lecture 06. Feb 2. (Th)

Lecture 07. Feb 7. (Tu) HW 1 due in class

Lecture 08. Feb 9. (Th)

Lecture 09. Feb 14. (Tu)

Lecture 10. Feb 16. (Th) HW 2 due in class

Lecture 11. Feb 21. (Tu)

Lecture 12. Feb 23. (Th)

Lecture 13. Feb 28. (Tu) HW 3 due in class

Lecture 14. Mar 2. (Th)

Lecture 15. Mar 7. (Tu)

Lecture 16. Mar 9. (Th) HW 4 due in class. *Final project topics decided*

**Spring recess**

Lecture 17. Mar 28. (Tu) HW 5 due in class

Lecture 18. Mar 30. (Th)

Lecture 19. Apr 4. (Tu) Project assignment 1 due in class

Lecture 20. Apr 6. (Th)

Lecture 21. Apr 11. (Tu)

Lecture 22. Apr 13. (Th)

Lecture 23. Apr 18. (Tu) HW 6 due in class

Lecture 24. Apr 20. (Th)

Lecture 25. Apr 25. (Tu) First draft of paper due in class

Lecture 26. Apr 27. (Th)

**Reading period**

May 9th (Tu) Final papers due 7pm