Arithmetic Geometry (18.782 Fall 2019)

Instructor: Junho Peter Whang
Email: jwhang [at] mit [dot] edu

Meeting time: TR 9:30-11 in Room 2-147
Office hours: M 10-12 or by appointment, in Room 2-238A

This is the course webpage for 18.782: Introduction to Arithmetic Geometry at MIT, taught in Fall 2019. This webpage will be updated regularly; be sure to refresh the page to access the most recent version.

Overview: Arithmetic geometry is an incredibly broad field of research at the interface of number theory and algebraic geometry. It has been particularly influential in the study of Diophantine equations (i.e. polynomials with integral coefficients, to be solved in integers or rationals). This course introduces some of the basic results and techniques in arithmetic geometry, with emphasis on Diophantine applications. Topics will include a subset of the following: the p-adic numbers, the local-global principle, algebraic geometry of curves, Mordell's theorem, Diophantine approximation, and Siegel's theorem.

Course policy and syllabus: The course policy and syllabus can be found here.

Prerequisites: 18.702 or equivalent (one year of abstract algebra)

Textbook: There is no required text; lecture notes and references will be provided. The following texts are excellent sources:

Notes for this course from previous years by B. Poonen (here) and by A. Sutherland (here) are also available.

Exams: There will be no midterm or final exam in this course.

Grading: The final grade will be based on the following scheme.


Problem sets

There will be 6 problem sets, each worth 10% of the final grade. Problem sets will be posted here. You should submit your problem set to the instructor by email in PDF form (LaTeX preferred) on the indicated due date before 11:59 pm.

Final project

Each student will write a short expository paper (approx. 5-10 pages) on a topic of their choice in arithmetic geometry, and give a short presentation (approx. 30 minutes) on the topic at the end of the semester. The collective goal is to highlight a variety of approaches to solving Diophantine equations developed in the early- to mid-twentieth century, many of which have formed the basis for current research. Some suggestions include, with varying levels of difficulty:

Students may also pursue other topics, provided that they fall within the scope of the course and are discussed with the instructor. The exposition should be aimed at fellow students learning the subject. Submitted articles will be compiled and made available to the class at the end of the semester.

Here are descriptions of some of the topics to help with choosing.

Lectures

Notes from the lectures will be uploaded here (last updated: December 1, 2019) as they are updated each weekend; be sure to clear the cache before reloading to get the most current version of the notes.


Go to the webpage of Junho Peter Whang