About
This is an undergraduate student seminar on geometry and topology. This semester, our main theme will be modular forms and their applications to geometry and topology. To begin with, in order to equip ourselves with basic notions and examples of modular forms, we will be working mostly algebraically and analytically from Serre's A course in arithmetic. But here are some motivations with geometric flavors (and beyond) just for you to skim through and keep in mind.
Here are some further references in company with Serre's and where we are heading later per interests of participants.
Here is a course description if you wish to register for credit. Note that this semester, due to the nature of the topics, you may not need topology or differential geometry as a prerequisite to understand the material to be covered per se. On the other hand, you may lack a certain amount of intuition/motivation from such experiences as we move on.
Talks (Thursdays 4:20–6:10 in College of Science Building M5024)
Sep 8, '22,
Kaicheng Dong,
Introduction to modular forms; finite fields and the quadratic reciprocity law
Sep 15, '22,
Rongming Yin,
Introduction to p-adic fields and the Classification of quadratic forms over the rationals
Sep 22, '22,
Rongming Yin,
Introduction to p-adic fields and the Classification of quadratic forms over the rationals, cont'd
Sep 29, '22,
Kaicheng Dong,
Introduction to Dirichlet L-functions
Oct 6, '22,
Yining Han,
Introduction to Dirichlet L-functions, cont'd & Dirichlet theorem
Oct 13, '22, Ziyin Hu, The modular group; modular functions
Oct 20, '22, Qiyuan Song, The space of modular forms; expansions at infinity
Oct 27, '22, Qiyuan Song, The space of modular forms; expansions at infinity (cont'd)
Nov 3, '22, Antong Zhu, Hecke operators; theta functions
Nov 17, '22, Antong Zhu, Hecke operators; theta functions (cont'd)
Nov 24, '22,
Hongxiang Zhao,
Introduction to elliptic curves, j-function, complex multiplication, and modular curves
Dec 1, '22, Hongxiang Zhao, Introduction to elliptic curves, j-function, complex multiplication, and modular curves (cont'd)
Dec 8, '22, Zeyang Ding, Introduction to Eisenstein Series and Differential Operators on Modular Forms
Dec 15, '22, Zeyang Ding, Introduction to Eisenstein Series and Differential Operators on Modular Forms (cont'd)