Undergraduate Seminars I: Fermat's last theorem

Fall 2024

This is the webpage for the section of Undergraduate Seminars I focused on Fermat's last theorem. A tentative syllabus can be found here.

Instructor: Avi Zeff

Time/place: Mondays 1 - 3 PM in 507 Mathematics

Homework

Homework 1
Homework 2

Other materials

Peer feedback form
Textbook
Project guidelines

Student projects

Mock Theta Functions and Ramanujan's Last Letter (Luca)
An Introduction to Elliptic Curves and their Cryptographic Applications (Kaitlyn)
Elliptic Curve Security (Zach K.)
Siegel Modular Forms (Zachary L.)
Making a Brief Acquaintance with Pell's Equation (Leo)
A Short Introduction to the Langlands Correspondence (Shiv)

Tentative schedule

Date Speaker Topic Sources Materials
September 16 Avi Introduction and logistics PDF
September 23 Leo Elementary approaches: infinite descent, n = 3, 4, sums of two squares [H, §1.4 - 1.7]
September 23 Avi Kummer's approach: proof for regular primes [H, §1.8] PDF
September 30 Luca Elliptic functions I [H, §2.1 - 2.5] PDF
September 30 Shiv Elliptic functions II [H, §2.6 - 2.12] PDF
October 7 Avi Number fields [H, §3] PDF
October 14 Zach K. Elliptic curves I: definitions and first properties [H, §4.1 - 4.5] PDF
October 14 Zachary L. Elliptic curves II: endomorphisms and bounds [H, §4.6 - 4.11] PDF
October 21 Kaitlyn Elliptic curves III: further properties and conjectures [H, §4.12 - 4.15] PDF
October 21 Zach K. Modular forms I: theta functions [H, §5.1 - 5.2] PDF
October 28 Zachary L. Modular forms II: definitions and first properties [H, §5.3 - 5.4] PDF
October 28 Shiv Modular forms III: operations [H, §5.5 - 5.8] PDF
November 11 Luca Modularity and Fermat's last theorem I [H, §5.8 - 6.1] PDF
November 11 Leo Modularity and Fermat's last theorem II [H, §6.2 - 6.7]
November 18 Kaitlyn Addenda and related problems [H, §6.7 - 6.8]
December 2 Avi Outline of the proof of modularity PDF
December 9 Students Project talks

[H] Hellegouarch, Yves. Invitation to the mathematics of Fermat–Wiles (Schneps, Leila, Trans.). Elsevier, 2001.