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A working seminar on Modular Forms

Fall 2021

Time & Location: All talks are on Wednesday at 2:00 PM unless otherwise noted.
Organizer: Victor H. Moll, Olivia Beckwith, and Vaishavi Sharma
 

Wednesday, October 6

Title: Modular forms as coefficients of the P-function

Victor H. Moll | Tulane University

Abstract: The study of polynomial equations is one of the basic problems in Number Theory. In this first talk we will show that “modular forms”, the subject of this seminar, appear in a natural manner in the study of the curve “quadratic in y = cubic in x”. The details are completely elementary.

Location: Gibson 310
Zoom access:
Time
: 2:00

 

Wednesday, October 13

Title: The First Appearance of Modular Forms

Victor H. Moll | Tulane University

Abstract: TBA

Location: Gibson 310
Zoom access:
Time
: 2:00

 

Wednesday, October 20

Title: First examples of modular forms: Eisenstein series

Olivia Beckwith | Tulane University

Abstract: Picking right up where we left off last week, we'll examine the Taylor coefficients of the Weierstrass P function in more detail and we'll compute their Fourier expansions. Then we'll define vector spaces which contain these functions and we'll show that they are finite dimensional.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, October 27

Title: Level One Modular Forms

Vaishavi Sharma | Tulane University

Abstract: In this meeting we'll show that the Eisenstein series G_k  studied in the previous lectures live in finite-dimensional vector spaces of functions on the upper half-plane with certain transformation properties. We will also use basic properties of these spaces to prove a surprising identity for the divisor sum functions sigma_k.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, November 3

Title: Modular forms on congruence subgroups

Olivia Beckwith  | Tulane University

Abstract: Last week we defined modular forms on the special linear group SL_2(Z). This time we'll replace SL_2(Z) with certain finite index subgroups. We'll also show how to apply these ideas to obtain a formula for the number of ways an integer can be written as a sum of four squares.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, November 10

Title: The j-function

Victor Moll | Tulane University

Abstract: We give a description of this important function and explain its role in connection to modular functions.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, November 17

Title: Modular functions and the inevitable j-function

Armin Straub | Tulane University

Abstract: We will review previous results on modular forms and the j-function, with the goal of (re)introducing the j-function as inevitable and natural.  We will discuss some of its properties as well as its fundamental role as a modular function, and might indicate extensions to the higher level case.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, November 24

Title: TBA

TBA | Tulane University

Abstract: TBA

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, December 1

Title: Algebraic relations between modular functions

Armin Straub | University of Southern Alabama

Abstract: We will continue the discussion of modular functions and show that any two are algebraically related.  This leads to the observation that modular functions like the j-function take algebraic values at quadratic irrationalities.  This is the tip of an impressive iceberg, and hints at the beginning of topics including complex multiplication and class field theory.

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, December 15

Title: TBA

TBA | Tulane University

Abstract: TBA

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

Wednesday, December 22

Title: TBA

TBA | Tulane University

Abstract: TBA

Location: Gibson 310
Zoom access: https://tulane.zoom.us/j/94370467366  Find Zoom information at the end of this webpage.
Time: 2:00

 

 

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