18.784 - Seminar in Number Theory - Spring 2008
The Mathematical Legacy of Ramanujan

| (Up) Dates | Homework | PARI | Further Reading | Handouts |

General Information

Time MWF, 2-3 p.m.
Location Building 2, Room 102
Professor Ben Brubaker (brubaker@math.mit.edu)
Office: 2-267
Office Phone: 3-4079
Office Hours: TBA, but always available by appointment.
TextbookNumber Theory in the Spirit of Ramanujan, by Bruce Berndt
Grade
Breakdown 		Homework -- 15%, In-Class Presentations -- 40%, Participation -- 10%, Final Paper -- 35%
Semester
Plan		 Class members will take turns in presenting small portions (3-4 page segments) of the book. Depending on our pace, we should have time left over to cover special topics in modular forms
Prereqs A first course in number theory, or at least some familiarity with these topics , would be quite useful. However, a mathematically mature student should be able to pick up the necessary background in number theory relatively quickly. In addition, many of the theorems we will discuss do not require (but are perhaps best stated in) the language of complex analysis, so some exposure to the subject is a plus. Students who have not had complex analysis may want to obtain a textbook on the subject for reference as they read along in Berndt. Ahlfors "Complex Analysis" is standard, though readers may find Levinson and Redheffer's "Complex Variables" a bit easier.

Date

Speaker

Material Covered
----FINAL PROJECTS ----
#31
    Friday, May 2
1. Daniel Briggs
2. Hyun Soo Kim
#32
    Monday, May 5
1. Charmaine Sia
2. Jon Hanselman
#33
    Wednesday, May 7
1. Geehoon Hong
2. Joseph Cooper
#34
    Friday, May 9
1. John Kim
2. Sungyoon Kim
#35
    Monday, May 12
1. Safia Chettih
2. Cotton Seed
#36
    Wednesday, May 14
1. Maria Monks
2. Connor McEntee
----ROUND 5 ----
#25
    Monday, Apr. 14
1. John Kim
2. Sungyoon Kim 		1. Dedekind Eta Function - Modularity
2. Eta function product expansion, second pf of mod.
#26
    Wednesday, Apr. 16
1. Cotton Seed
2. Joseph Cooper 		1. Hecke Operators - definition of T_n
2. Action of T_n on modular forms
#27
    Friday, Apr. 18
1. Geehoon Hong
2. Hyun Soo Kim 		1. Eigenfunctions of Hecke operators
2. Examples of L-functions
!
    Monday, Apr. 25
No Class - PATRIOT'S DAY
#28
    Friday, Apr. 25
1. Jon Hanselman
2. Daniel Briggs 		1. Theta functions and quadratic forms
2. Theta functions as modular forms
#29
    Monday, Apr. 28
1. Connor McEntee
2. Safia Chettih 		1. Spaces of Integral Weight modular forms
2. Congruence subgroups and modular forms
#30
    Wednesday, Apr. 30
1. Maria Monks
2. Charmaine Sia 		1. 1/2-integral weight modular forms
2. Generalizations of modular forms
----ROUND 4 ----
#19
    Monday, Mar. 31
1. John Kim
2. Jon Hanselman 		1. Duplication and Dimidation (5.3, Thms 5.4.1-4)
2. P,Q,R formulas (Thm 5.4.6 to end of 5.4)
#20
    Wednesday, Apr. 2
1. Maria Monks
2. Safia Chettih 		1. Elliptic Functions I (unimodular transformations)
2. Elliptic Functions II (bases, basic properties)
#21
    Friday, Apr. 4
1. Charmaine Sia
2. Connor McEntee 		1. Elliptic Functions III (Weierstrass p-function)
2. Elliptic Functions IV (ODE for Weierstrass p)
#22
    Monday, Apr. 7
1. Hyun Soo Kim
2. Geehoon Hong 		1. Sums of squares, take 2 (6.2 thru Thm 6.2.6)
Sums of trian. nbrs, "taxicab" take 2 (finish 6.2)
#23
    Wednesday, Apr. 9
1. Joseph Cooper
2. Sungyoon Kim 		1. Modular Equations (6.3, thru pf of Thm 6.3.2)
2. Summary of deg. 3 modular equations (finish 6.3)
#24
    Friday, Apr. 11
1. Daniel Briggs
2. Cotton Seed 		1. Modular Forms I (modular gp, mod forms)
2. Modular Forms II (Eisenstein series, take 2)
----ROUND 3 ----
#13
    Monday, Mar. 10
1. Charmaine Sia
2. John Kim 		1. Repns of quadratic forms (3.7, thru 3.7.5)
2. x^2+xy+y^2 (Lemma 3.7.6 to end of section)
#14
    Wednesday, Mar. 12
1. Connor McEntee
2. Cotton Seed 		1. Notes on 3.8 - general thms about sums of sqs.
2. Bernouilli Numbers, Eis. series (section 4.1)
#15
    Friday, Mar. 14
1. Sungyoon Kim
2. Safia Chettih 		1. Recurrence for Eis. Series (4.2, thru Thm. 4.2.3)
2. A second recurrence (Thm 4.2.4, finish Sect. 4.2)
#16
    Monday, Mar. 17
1. Hyun Soo Kim
2. Daniel Briggs 		1. New Proofs of Partition Congruences (4.4)
Notes on Chapter 4 (section 4.5)
#17
    Wednesday, Mar. 19
1. Jon Hanselman
2. Joseph Cooper 		1. Intro to hypergeometric funs. (5.1, thru Exer. 5.1.5)
2. Landen's Transformation (finish 5.1)
#18
    Friday, Mar. 21
1. Maria Monks
2. Geehoon Hong 		1. Phi^2(q) as a hypergeom. fun. (5.2, thru 5.2.5)
2. Finish the proof of theorem 2.5.8
----ROUND 2 ----
#7
    Monday, Feb. 25
1. Sungyoon Kim
2. Daniel Briggs 		1. Generating functions for p(7n+5), cong. mod 49 (finish 2.4)
2. Literature review on parity of p(n) (p. 43, 44)
#8
    Wednesday, Feb. 27
1. Hyun Soo Kim
2. Safia Chettih 		1. Preliminaries to proving Thm. 2.5.1 (p. 45-46)
2. Proof of Theorem 2.5.1
#9
    Friday, Feb. 29
1. Connor McEntee
2. Maria Monks 		1. Pf. of Theorem 2.5.2, Notes through p. 51
2. Conjectured properties of tau (Notes p. 52, 53)
#10
    Monday, Mar. 3
1. Charmaine Sia
2. John Kim 		1. Proofs on sums of two squares (Section 3.2)
2. Sums of four squares (Theorem 3.3.1 and 1st proof)
#11
    Wednesday, Mar. 5
1. Jon Hanselman
2. Cotton Seed 		1. Sums of six squares (Section 3.4)
2. Sums of eight squares (Lemma 3.5.1, Thm 3.5.2)
#12
    Friday, Mar. 7
1. Joey Cooper
2. Gehoon Hong 		1. Sums of eight squares (Thms. 3.5.3,3.5.4)
2. Sums of triangular numbers (Section 3.6)
----ROUND 1 ----
#1
    Monday, Feb. 11
1. John Kim
2. Daniel Briggs 		1. Section 1.2 - What are q-series and theta functions?
2. q-analogue of binomial thm. (up to middle of p. 10)
#2
    Wednesday, Feb. 13
1. Connor McEntee
2. Maria Monks 		1. Jacobi Triple Product Identity (thru Cor. 1.3.5)
2. Combinatorial identities through exercise 1.3.11
#3
    Friday, Feb. 15
1. Cotton Seed
2. Joseph Cooper 		1. Ramanujan's Psi Summation (thru Thm. 1.3.12)
2. The Quintuple product identity (thru Thm. 1.3.19)
#4
    Tuesday, Feb. 19
1. Safia Chettih
2. Hyun Soo Kim 		1. Corollaries of Thm. 1.3.19 (up through end of 1.3)
2. 1.4 Notes, focusing on ecercise 1.4.1
#5
    Wednesday, Feb. 20
1. Jon Hanselman
2. Sungyoon Kim 		1. Congruences for Tau (Section 2.2)
2. Partition congruence (first two proofs, thru top of p. 34)
#6
    Friday, Feb. 22
1. Charmaine Sia
2. Ben Brubaker 		1. Third proof of partition congruence (Theorem 2.3.4)
2. Partition congruences mod 25 and mod 7 (thru pf. of Thm. 2.4.1)

Homework Assignments

  • Here is a bonus homework assignment on numbers of solutions to certain quadratic forms.

  • As you'll notice in Berndt's book, there are many exercises embedded in the text. Homework is meant only to make sure you're keeping up with the class (both in a timely fashion and in understanding the material). Most often, your homework will always be to read the section to be discussed in class and pick 2 or more of the exercises in the included text, write down the solutions, and turn it in at the beginning of the next class.

  • You should just make sure to turn in problems at least once a week, and to have completed at least 30 problems by the end of the semester.

    • Announcements & Dates

    • Important Dates
      • Wednesday, February 6: First Day of Class
      • Monday, February 18: President's Day -- No Class
      • Tuesday, February 19: CLASS HELD -- MONDAY SCHEDULE
      • Friday, March 7: ADD Date
      • Mon-Fri, Mar 24-28: Spring Break
      • Monday, April 21: Patriots Day
      • Thursday, April 24: DROP Date
      • Wednesday, May 14: Last Day of Class
      • Mon-Fri, May 19-23: Final Exam Week

    PARI

    PARI is a computer algebra system written especially for number theory computations. You may want to use it to test conjectures and do experiments with larger numbers. It's also a great way to double check your homework. It is FREE, and can be downloaded at the following main site:

  • PARI/GP Main Page (click on the "Download" link on the left margin)
    • Here are some instructions for downloading the files from the above site using UNIX:
  • Unix Installation Instructions

    • If you have a Windows-based computer, then the installation should be automatic upon downloading the appropriate file from the PARI/GP page (though I haven't tried it). For Mac users, it may be easiest to download the program Fink and Fink Commander from the web first. These programs fetch all of the appropriate files for you and then compile them on your hard drive. You can then run PARI from the terminal window (in OS X).

    Here is some additional documentation:

    You can also install PARI using your personal Athena file space, decompressing the downloaded file in /mit/username. If your personal space is full, we can even request a course locker for 18.784 with a gigabyte of space and place it there.

    Other Computing Options

  • If you have other computer-algebra software that can do the same computations, such as Mathematica or Magma, then you may use it for the course. However, PARI is the language I will use in in-class examples, so you should probably have some minimal programming experience in order to write and de-bug programs in your chosen computer-algebra system.
  • Sage is another free computer algebra system started by William Stein, now at Univ. of Washington. It includes a PARI interface (as well as many other programs). You can download it from the site above or even use an online notebook.

    • Further Reading

  • Here's an article by Berndt and George Andrews that recently appeared in the Notices of the AMS Eventually, I'll put up some additional things to read.

    • Solutions and Handouts:

  • Notes for Wednesday's final project are now part of "Spaces of Modular Forms"

  • Spaces of Modular Forms, notes for Monday and Wednesday classes, including 1/2 integral weight modular forms, last updated Wednesday, April 30, at 2:00 pm.

  • Theta functions, notes for projects beginning this coming Friday, last updated at 1 p.m. on Tuesday, April 22

  • Hecke Theory (a continuation of the modular forms notes, last updated at 11:30 pm, Tuesday, April 15)

  • Modular Forms (last updated Saturday, April 12, 1 pm)

  • Elliptic Functions (last updated Friday afternoon, 5/28) and the LaTeX file I used to produce this output: elliptic.tex

  • Final Project Proposal Guidlines

  • Eventually, I may post homework solutions and in-class handouts here.