Math 490, Spring 2012

Curl (Collaborative Undergraduate Research Lab)


Class Room:	Wed: VAN VLECK B337
	Mon, Fri: VAN VLECK B107
Class Time:	MWF 11:00AM - 11:50AM
Final Paper Due:	May 18, Noon
Instructor:	Melanie Matchett Wood
Instructor's Office:	315 Van Vleck
TA:	Lalit Jain
Office Hours:	Prof. Wood: Mon 10-11AM, Fri 1:30-2:20PM
Computer Lab Office Hours:	Lalit Jain, Van Vleck B107, Tues 4-5PM, Wed 3:30PM

Application

Enrollment in this course requires consent of the instructor. If you are interested, please apply by completing the application avilable here.

An introduction to mathematical research

Algebraic curves over finite fields are beautiful and subtle objects at the nexus of number theory and algebraic geometry. They can also be defined simply in terms of polynomial equations whose coefficents are in a finite field, such as the integers modulo a prime. They are also typically not the topics of undergraduate courses.

This course is both an opportunity to get aquainted with algebraic curves over finite fields and an introduction to mathematical research. For the first third of the course we will learn the background material. There will not be traditional lectures, but rather presentations by the lecturer interspersed with working through examples in groups. Participation is crucial to your success in the course. During this part of the course there will be quizzes and homework problems (including some which require the use of the computer), and the homework solutions will need to be written up using the mathematical typsetting language LaTeX. The remainder of the course will be focused on the research projects -- I will assign research projects, to groups of 2-3 students, related to these topics, and our class meetings will consist of reports from the groups, with the occasional presentations from the lecturer when questions come up. The projects will be largely computational (meaning using the computer a lot!) and will be exploring questions at the frontier of current mathematical knowledge. The last two weeks will be devoted to working on the final papers, which will give a full report of the research done by each group. There is no required textbook, so students' notes from the presentations will be critical as a reference.

I will have a small number of summer research intern positions in Summer 2012 for students from the course to continue their research program with me.

Goals of the course

learn some basic background about algebraic curves over finite fields, and more detailed information about specific families of curves
learn the process of producing mathematical data and exploring it for patterns
develop mathematical computer programming skills
develop presentation skills
develop mathematical writing skills
develop LaTeX proficiency

Prerequisites

No computational experience is necessary, but interest in exploring mathematics through computer generated examples is a must!
Linear algebra will be necessary.
A knowledge of number theory and/or abstract algebra (e.g., finite fields) is a plus, but one can also learn what is needed along with the course material.

Some recommended reading

Here's a bit of mathematics to begin learning or review 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.

Algebra -- group, ring, field, homomorphism, finite fields, Z/nZ, ideal, polynomial ring, some references:

Abstract Algebra by Dummit and Foote;
Contemporary Abstract Algebra by Joseph Gallian
Your favorite abstract algebra book

Course requirements

Math 490 will require a lot of work, at least as much as other advanced math classes, but also requiring much more initiative and independent work. The class is a serious commitment, but the payoff for you will be getting a glimpse of mathematical research and some really interesting open (unsolved) questions in mathematics. Please only consider this class if you will have time in your schedule for a serious commitment to a research project.

Caveat : We will be doing mathematical research, and by it very nature, we can't predict the outcome and so it is possible that the course requirements may change to better reach our resaerch goals, e.g. perhaps groups will be merged or split if it turns out two questions have a common thread or one question splits into two, or perhaps presentations will be longer and less frequent if we need more time to devote to each presentation.

Preparation: First 5 weeks (Jan 23-Feb 24)

Show up on time and bring paper and writing implement to every class Not only will your notes be a crucial reference (as there is no textbook), but you will also need these for quizzes (see below).
Quizes (10%) There will be 5 minute quizes at the start of many classes in the first 5 weeks, which will require basic recall of definitions from previous classes, so you will need to review your notes before each class.
Homework (10%) Homework will be given during each class period, and due the following Friday in class. For example, problems given Monday, Wednesday, and Friday, January 23, 25, and 27 are due at the start of class Friday, February 3. Homework must be typset in LaTeX (see below), and a printed copy of your pdf or dvi is due in class. The final homework assignment is due March 2. Students may work together on problems but must each produce their own latex file of written solutions, and should include in their write-up the names of each student they worked with.
Computer lab office hours (10%) Each student should attend computer lab office hours with Lalit Jain once each week to keep on track with the LaTeX and compuational requirements.
Project Work The projects will be introduced during weeks 4 and 5 and groups should start working on them as soon as their project is introduced.

Projects: February 27-April 27

Project Work The project work should include from each group member at least 5-8 hours of work per week.
Presentations (30%) Each group will give a weekly project report. This report should include a handout or projected computer presentation which gives relevant charts or tables of data, and summarizes (e.g. in clearly written bullet points) the main updates of the report, as well as future questions to be considered. The group may decide how to split the presentation duties, but it is recommendation that they rotate in a reasonable way. Group 1 will give a presentation each Monday, groups 2 and 3 each Wednesday, and groups 4 and 5 each Friday. The presentations will be graded based on the expectations for project work given above.
Presentation Paricipation (10%) Students will be expected to given constructive feedback, ask questions, and make suggestions on the other groups' presentations.
Computer lab office hours Each student should attend computer lab office hours with Lalit Jain once each week to keep on track with the compuational requirements.

Final Papers: April 30-May 11

Paper outline (10%) An outline of the final paper for each group is due to be emailed to the class email list by 10:30am April 30.
Paper drafts (10%) Each group will share two (partial) drafts of their final paper with the class for constructive feedback and will turn in one full draft to the TA. The (partial) drafts will be due emailed to the class list by 10:30am on the following dates: May 2 (groups A and B), May 4 (groups C and D), May 7 (groups E and A), May 9 (groups B and C), May 11 (groups D and E).
Final Paper (10%) The .tex and all necessary files, along with a pdf, is due emailed to the instructor by noon on May 18.

Grades

Grades will be based on your effort in all of the above, in the given percentages. Historically most students (the students who worked very hard) in Math 490 get a A, and most of the rest get an AB.

LaTex Resources

You'll want to download a LaTeX editor for writing up your assignments and the eventual write up of your project. Please set this up before the semester begins.

Windows: Download MikTeX.
Mac: You can download either of the following: TeXShop or MacTeX (There seems to be some problem with getting the whole TexShop package to work; you might want to download just the basic program- start with the BasicTeX.dmg file here, and then download the proper update from the TexShop link.)
Linux: If you use Linux, you might already have a TeX editor installed somewhere. If not, I think TeXLive might work.
Any platform: One option is to use Auctex, which runs under emacs, and Skim as a viewer (this one is Mac only). One nice emacs client for a mac is Aquamacs, which, I think, comes with AucTex preinstalled.
Yvonne Nagel has a pretty comprehensive list of resources here.
Finally, here is a sample LaTeX file to get you started.

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.

The Not So Short Introduction to LaTeX (pdf)
Googling a specific TeX question usually works out.
LaTeX Symbols
A very comprehensive Symbol list.
Detexify -- draw a symbol and detexify tries to figure out how to produce it with LaTeX.
If Googling doesn't work, go to the stackexchange Q&A site to have your questions quickly answered by experts.
Beamer quickstart -- a guide to making slides in LaTeX.

Computers

Lalit Jain has created a nice Sage/Linux reference which is available here. His other computing documents are available here. I like this Sage reference manual

Math department linux machines you can work on are listed here.