Lecture: MWF 3:00-3:50, Eng E206.

Prerequisite: Math 566 & 519 or permission of professor.

Course description: In this course, we will focus on the topics of abelian varieties and theta functions. We will begin with the work of Abel and Jacobi: while studying arclength and elliptic integrals, they developed a connection between algebraic cubic equations and complex tori, which became the backbone in the study of elliptic curves. In the second part of the course, we will cover theta functions, periodic functions that are needed to study abelian varieties (a higher-dimensional version of elliptic curves) and Jacobians of curves. In the last part of the course, we will study group actions on the upper half plane, modular curves, moduli spaces for abelian varieties and polarizations.

The material in this course has deep connections to number theory, algebraic geometry, complex analysis, hyperbolic geometry, and topology. It also has applications to physics, dynamical systems, and cryptography. Towards the end of the course, everyone will focus on a topic for a project which is especially interesting to them.

Syllabus: syllabus

Texts: Many excellent textbooks contain this material. We will focus on TBA

Grading: This is an advanced graduate course and course grades will be computed accordingly.
30% Homework; 20% computer lab; 20% Presentation; 20% Project; 10% Discussion sections.

Homework and short presentation: Homework is the most important part of this class. It should demonstrate your knowledge of the material, your investigation of open ended questions, and your skill at writing proofs. Please make sure your homework is neat, legible, and stapled. I encourage you to brainstorm the problems together and write up your solutions independently.
HW1
HW2

Project: The project is an opportunity to learn more about a topic in this area that interests you or will be relevant for your future graduate work. It gives us a chance to hear about fascinating ideas which we will not have time to cover in class. It is also a good opportunity to develop more skill at writing and speaking on mathematics. Detailed information on projects

Help: Help is always available if you have trouble with homework or lecture material. If your classmates can't answer your question, come ask me! Office hours will be (TBA) or are available by appointment.