Math 490, Spring 2011

Curl (Collaborative Undergraduate Research Lab)

Problem sets
(Some) Lecture notes
Details of the class
Latex Resources
Sage Resources
Other Resources
Lecture Room: 2124 Chamberlin
Lecture Time: MWF 1:20-2:10
Final Exam: None
Lecturer:David Brown
Office: 319 Van Vleck
Phone: (510)508-0255

Office Hours: David: MW 5-6
Mon. 2/7, OH moved to 4-5
Lalit: Tu 4-5,Th 5-6 in B107

An introduction to mathematical research

Elliptic curves, modular forms, and Galois representations are singularly central objects of mathematical study and are the topics of many cornerstone results in number theory and of many open conjectures. They are also typically not the topics of undergraduate courses.

This course is both an opportunity to get aquainted with elliptic curves, modular forms, and Galois representations and a gentle introduction to mathematical research. For the first third of the course I will lecture about these topics and assign interesting but nonstandard homework problems (including many which require the use of the computer -- an excellent way to build an intuition for these mathematical objects). The majority of the course will be more informal -- I will assign research projects, to groups of 2-3 students, related to these topics, and our class meetings will be more of a discussion of results, with the occasional impromptu lecture when questions come up.

The goal is that everyone will leave with a taste of mathematical research and an appreciation for modular forms and related topics.

Suggested Texts

There is no required text, and indeed no single text which covers everything that I'd like to cover. One article which covers much, but not all, of the minicourse is which is available here on Google books; alternatively, I have put a pdf here. You aren't required to read this, but if you're looking for more detail than your notes from lecture, this article is a good start.

Another good (and free) reference is which is available here; the last chapter has a nice discussion of elliptic curves.

A nice list of resources, some of which are freely available online is here.

Sample projects

(Disclaimer to potential students: these sample projects shouldn't necessairly make sense to you yet; that will be the goal of the course!)


Class Organization

LaTex Resources

You'll want to download a LaTeX editor for writing up your assignments, the eventual write up of your project, and for making slides for your final presentation. Please set this up before the semester begins. Next, it's important to have somewhere to turn when you've got an error that WON'T GO AWAY, or when you can't remember what the heck that symbol was.

Mathematical software

Many of the projects will make extensive use of the mathematical software Sage. The following computer labs are available for use:

Other Mathematical resources

Recommended reading

Here's a bit of mathematics to begin learning before the semester begins. I've included links to wikipedia pages which contain definitions and examples; in addition pick up almost any undergraduate or graduate book on the subject and read a bit, or google around and find notes if you are not close to a library. I've suggested a few books or notes to read. I absolutely do not expect people to master all (or even most) of the material in these references; please do however at least familiarize yourselves with the basic definitions and examples of these objects

Teaching Assistant

Lalit Jain

Lalit has created a nice Sage/Linux reference which is available here.