An introduction to mathematical research

Suggested Texts

• "Elliptic curves, Modular Forms, and Applications" by Joe Buhler,

• "Elementary Number Theory: Primes, Congruences, and Secrets" by William Stein (creator of Sage),

Sample projects

• Make a comprehensive database of slopes of modular forms and test various conjectures (e.g., use recent massive advances in mathematical computation to update Gouvea's 2000 paper "Where the slopes are").
• Compute lots of things about congruences of modular forms.
  • Test the Frey-Mazur conjecture (this has applications to solving generalized Fermat equations, i.e. equations of the form x^a + y^b = z^c).
  • Test variants and generlizations of Szpiro's conjecture.
• (Partially) Compute the possible images of the mod n Galois representations associated to elliptic curves over Q for small values of n.
• More details and projects to come! The write ups of each project will eventually be posted here.

Prerequisites

• No computational experience is necessary.
• Linear algebra and facility with power series will be necessary.
• A bit of knowledge of abstract algebra (e.g., group theory and finite fields) and of complex analysis will be useful, but one should be able to learn what is needed on the fly.

Class Organization

• Minicourse. The first third of the semester will be a short introduction to elliptic curves, modular forms, and Galois representations.
• Problem Sets. I will assign problem sets of 2-4 problems every class (roughly 6 per week; there is a lot to learn!). Group work is required; see directly below. Problem sets must be typed up in LaTeX format (see below) for full credit. Go here to find the problem sets.
• Group Problems. I will organize you into groups of three or four, to work together in and outside of class. Each assignment will be written in LaTeX (see below) and will be due two classes later at the beginning of class. Group problem sets must be typed in LaTeX for full credit, and will not be accepted late.
• Research Problems. This will be the bulk of the class (more later!). There will be no official homework after the first third of the class; hopefully though everyone will be so excited by their projects that they need no external motivation!
• Mid semester Presentations. Research groups will periodically report to the class about their progress, as well as give an initial introduction of their project to the class.
• Final Presentations. At the end of the semester, each group will give a 20 minute presentation (with slides made in LaTeX) to the class (and whoever else decides to attend!).
• Final paper. The culmination of the class will, hopefully, be original research! I will have everyone write up their results in a professional manner. These will all be posted on the arXiv, and some will be submitted for publication.
• Exams. None.
• Grades. Everyone should get an A. If you slack off a lot, you will get a B or an AB.

LaTex Resources

• 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: I use Auctex, which runs under emacs, and Skim as my 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.

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

Mathematical software

• Download Sage here.
• Play around with it a bit:
• Learn how to factor an integer.
• Learn how to factor a polynomial.
• Find as many solutions as you can to the equation x^2 + y^3 = z^7
• Find as many coprime integer solutions as you can to the equation x^2 + y^3 = z^7 (i.e., solutions such that x,y, and z have no common prime divisors)
• Awesome forum for Sage questions (don't be intimidated! ask away!).

• Van Vleck 101 -- This will be available 9am-4pm, M-F.
• Van Vleck B107 -- This room requires a reservation. I have reserved it for two hours a week, during which Lalit will be available to help you with Sage and Latex.
  • Week one hours: Thursday, January 20, 4-6 pm
  • Hours the following weeks: Tu 4-5, Th 5-6

Other Mathematical resources

• MathSciNet: This is the definitive database of published math papers and books, and often contains links to online copies of papers (when an online copy exists). Also key: it contains reviews of papers, and lets you do fun things like search for all papers that cite a given paper (I find both of these very useful when reading a paper).
• The arXiv: Once a research paper is accepted for publication, it can take 2-3 years before it is actually published (!). Moreover it can take 3-6 months (or so..) from submission of a paper before a decision is made about whether a given journal will publish it. The arXiv is a preprint server -- i.e., it is a place where papers live during the pre-publication limbo.
• Not every author puts their papers on the arXiv; check an author's web page to see if preprints are posted there
• Google books.
• Google scholar.
• There are more illicit means of obtaining books, which I will not advertise here.

Recommended reading

• Algebra -- group, ring, field, homomorphism, examples: R (real numbers), C (complex numbers), the symmetric group, finite fields, Z/nZ, matrix rings, ring of formal power series, structure theorem of finitely generated abelian groups; some references:
• Abstract Algebra by Dummit and Foote;
• Contemporary Abstract Algebra by Joseph Gallian
• Number theory --
  • Basic -- congruences and modular arithmetic, proving that equations have no integer solutions. Reference: AoPS notes
  • Advanced -- Z_p (p-adic integers), profinite groups and their topology, number field; references:
    • Algebraic Number Theory by Neurkirch
    • Nigel Boston's notes
    • More notes on algebraic number theory.
• Complex Analysis -- holomorphic functions, the exponential function and its power series expansion, complex line integrals, residue of a holomorphic function; some references:
• Basic Complex Analysis by Marsden and Hoffman
• Course at MIT
• Notes by Frank Neubrande
• Representation Theory -- Learn the definition of a representation of a finite group. Know a few examples (see e.g., Section 3 of the reference below). Browse around the notes and get a flavor of the theorems of the subject.
• Notes

Teaching Assistant