Organizer: Austin Lei
Modular forms, put simply, are holomorphic functions that satisfy a lot of "symmetries". They appear plentifully in number theory, with applications to algebraic geometry, representation theory, algebraic topology, and other fields. Famously, modular forms were an important component in the proof of Fermat's Last Theorem.
Modular forms can be studied with both algebraic and analytic methods, and depending on student interest, we explore modular forms via many different perspectives. We also may spend time discussing some interesting applications of modular forms; for example, their use in the proof of higher-dimensional sphere packing, and counting the number of ways to represent and integer as a sum of four squares.
Background in algebra will be helpful, especially a background in group theory. Some background in complex analysis and Fourier analysis will also be helpful, although the important details will be covered along the way.
References
Here is a list of potential references that could be covered throughout the semester:
Format/Grading
Students will sign up to present for 1 hour time slots (45 minutes of talking with 15 minutes for discussion). The list of topics to sign up for can be found here.
While I will give an outline for topics to discuss, there will be some sections left free for students to talk about topics of their own choosing, provided they are related to modular forms. Students should expect to present roughly 3 times in the semester.
Students are required to schedule a meeting with me before their talk to work out any questions they have and to nail down what will be happening during the talk.
Grading will be determined by attendance, giving talks, and providing feedback on talks. Taking notes for others' talks or providing notes for one's own talk is not required, but can help improve one's grade. More specific details can be found in the notes for the first talk.
Schedule
We will meet Mondays from 4:30-5:30 pm in Math 622, and Wednesdays from 4:00-5:00 pm in Lewisohn 610.
| Date | Speaker | Abstract | References | Notes |
|---|---|---|---|---|
| 9/8 | Austin Lei | Organization + Introduction to Modular Forms: We will discuss the historical context behind modular forms, some interesting applications of modular forms, as well as brief overview of potential topics to discuss throughout the course. | [1] | notes |
| 9/12 | Austin Lei | Fundamental Domain + Fourier Expansions of Modular Forms: We will define a fundamental domain for the $\mathrm{SL}_2(\mathbb{Z})$ action on the upper half-plane $\mathbb{H}$. We will then spend time talking about the important elements of Fourier analysis needed for the class, before talking about the Fourier expansions of modular forms. | [1], [2] | notes |
| 9/15 | Yifan | Complex Analysis Review, Riemann Surfaces | [2] | notes |
| 9/17 | Andrew | Complex Analysis Review, Classical Riemann Surfaces, $(\Gamma \backslash \mathbb{H})^*$ as a Riemann Surface | [2] | notes |
| 9/22 | Jasper | Elliptic Functions, Lattice Functions, and Eisenstein Series | [1], [2] | notes |
| 9/24 | Andrew | Elliptic Functions, Lattice Functions, and Eisenstein Series | [1], [2] | notes |
| 9/29 | Andrew | Zeros of Modular Forms | [1] | notes |
| 10/1 | Nikhil | The algebra of modular forms and the $j$-invariant | [1], [2] | notes |
| 10/6 | Jasper | Fourier expansions of Eisenstein series and bounds on Fourier coefficients of modular forms | [1], [2] | |
| 10/8 | Jarrett | Fourier expansions of Eisenstein series and bounds on Fourier coefficients of modular forms | [1], [2] |