Math 788G: The geometry of numbers.

Learning Outcomes: The successful student will develop an understanding of and appreciation for beautiful work of Gauss, Dirichlet, Minkowski, Hermite, Siegel, Davenport, Bhargava, and many others, which solves number theory problems by formulating them in terms of counting lattice points.

This is a highly suitable topic for Ph.D. dissertations.

Frank Thorne, LeConte 400G.

Office hours: Tuesdays, 2:00-5:00 (subject to change).

Comprehensive exam syllabus. (to be done)


For inspiration, read these papers by Manjul Bhargava, Wei Ho, Arul Shankar, Melanie Matchett Wood, among many others.

Notes on quadratic forms, by Andrew Granville.

Gauss' class number problem for imaginary quadratic fields, by Dorian Goldfeld.

On a principle of Lipschitz (a.k.a. Davenport's lemma), by Harold Davenport.

Geometry of numbers with applications to number theory, by Pete Clark.

Bhargava, Shankar, Tsimerman on Delone-Faddeev, Davenport-Heilbronn, and secondary terms.

On the class number of binary cubic forms, by Harold Davenport.

Course notes:

Lectures 1-3: introduction; counting lattice points; binary quadratic forms.

Lectures 4-10 + 1: The theory of binary quadratic forms.

Lectures 11-17: Real quadratic forms; Davenport's lemma; introduction to lattices.

Lectures 18-21: More on lattices, applications of Minkowski's theorem.

Lectures 22-28: Binary cubic forms and the Delone-Faddeev correspondence.

Lectures 29-32: The rest of the course.

Extra (Tuesday) lectures.

Homework assignments:

Homework 1, due Monday, January 27.

Homework 2, due whenever, but please don't give me a huge stack of homeworks the first week of May...

Homework 3.

Homework 4.