• Understand techniques used by number theorists to count lattice points.

• Understand examples of lattice point parametrizations of arithmetic objects.

• Understand and appreciate classical works of Gauss, Dirichlet, Siegel, Minkowski, and others.

• Be introduced to contemporary and ongoing works of Bhargava, Shankar, Wood, among many others, and lay a foundation for further study.

• Grow technical skills in algebraic and analytic number theory.

• Solve problems related to all of the above.

Makeup dates or times will be discussed with the class, and scheduled at a time when everyone can make it.

Homework (90%) : Homework assignments will be assigned periodically. Discussion sessions will be organized if there is sufficient student interest.

Seminar Reports (10%) : Students are expected to attend research talks relevant to their interests. Opportunities include the Algebra, Geometry, and Number Theory Seminar, the Department Colloquium, and the Palmetto Number Theory Series (PANTS), among others.

(Students wanting to travel to PANTS will have their travel expenses covered; ask me for details.)

Students should write up "responses" to at least four of the talks, of at least half a page each. These might summarize the talk, go into detail about a technical point you found interesting, describe questions you had from the talk (and answers, if you learned them on your own afterwards), outline research questions you thought of related to the talk, etc. Just respond to the talk in some mathematically serious way.

You are also required to fill out an "attendance verification quiz" on Blackboard, by university policy. (Details will be announced.)

Students are guaranteed an A for 75%+, B for 50%+, C for 40%+, D for 30%+.

There will be no exams.

Students who have passed their comprehensive exams may be excused from the homework (but not the seminar reports) at their advisor's discretion: please ask your advisor to send me an email, if he or she believes that your priorities should lie elsewhere.

• Academic honesty is expected of all students. Collaboration on homework is encouraged, but don't copy anyone else's writeup.

• Roll will not be taken.

• Late homeworks will be: always accepted in case of illness, emergency, or university-sanctioned absence; generally accepted in case of other legitimate excuses, such as a scientific conference; and may be accepted otherwise at the instructor's discretion. Please ask in advance.

• If you have difficulty seeing the board or hearing the lectures, or a related problem, let me know ASAP, and I will do something about it.

• Harassment, bullying, racism, sexism, and homophobia will not be tolerated. Please bring any incidents which I don't notice to my attention.

• If you require disability-related accommodations, please talk to the Student Disability Resource Center as soon as possible. It is your responsibility to advise me of any needed accomodations. (Note that there will not be any work assigned which is subject to a time limit.)

• Please contact me if you have any questions about the course, about my expectations, about my lectures, about the homeworks, about the reading, or about anything else. The subject is demanding and it is my job to help you succeed. The best ways to get help are to come to office hours (no appointment necessary) or to e-mail me (during the week, I will almost always reply within 24 hours). If neither of these work for you then please e-mail me to set up an appointment.

Highly subject to change, and feedback is welcome. Roughly speaking, the plan is to cover the following topics:

• The Circle and Hyperbola Problems: 2 weeks

  How do you count lattice points inside circles and hyperbolas? These questions anticipate many of the technical tools of the trade, and we will investigate several methods.

• Binary Quadratic Forms: 4 weeks

  We will thoroughly investigate questions involving binary quadratic forms, including parametrization theorems and "composition laws", as well as counting problems.

• Dirichlet's Class Number Formula: 2 weeks

  Dirichlet's Class Number Formula is a beautiful classical theorem, the proof of which puts the above ideas to good use, and which will be given in full.

• An Overview of Arithmetic Statistics: 1 week

  What is arithmetic statistics about? We will give an overview.

• The Davenport-Heilbronn Theorem: 2 weeks

  The Davenport-Heilbronn Theorem is an important landmark in arithmetic statistics, and an excellent model for many of the later works that were to follow.

  We will not give a complete proof, but we will introduce many of the relevant ideas.

• Contemporary Work: 3 weeks

  We will conclude with a discussion of some recent or ongoing works in the field. Several possible topics will be suggested, and we will decide together where to focus.

• Homework 1, due Friday, September 20.

• Part 1: Intro to lattice point counting and binary quadratic forms.

• Part 2: More on binary quadratic forms.

• Part 3: More on binary quadratic forms; Dirichlet's class number formula (definite case).