Summer Undergraduate Research in Number Theory at Amherst 2015

Description: Amherst College undergraduate students are invited to apply to spend 8 weeks of summer 2015 working together as a small group under the supervision of Prof. Folsom on an original research project in pure mathematics, in the area of number theory. See topic section below for more details. Approximately three students will participate.

Dates: June 16 - August 8, 2015. Participants are required to be in residence for the entire duration of the program. Minor exceptions may (or may not) be permitted; these should be discussed in advance. During the program, participants will also travel to and apply to give a talk on their summer research at MAA Mathfest in Washington, DC, August 5-8, 2015. (Funding will be provided.)

Funding: Participants will receive a stipend of $3680, divided into multiple payments over the course of the program. Participants may use a portion of this funding as “AC dollars” to pay for daily meals in the Amherst College dining hall. $672 total will fund 3 meals per day for 8 weeks; the remaining ~$3008 is an additional stipend. Funding is provided by NSF CAREER Grant DMS-1449679.

Housing: Participants will additionally receive (at no cost to them) on-campus housing, subject to availability after formally applying to Amherst College Housing.

Prerequisites: There are no formal prerequisites, however, the project will be most accessible to students who have taken (or are currently taking) at least one of Math 350 (Groups, Rings and Fields), Math 355 (Analysis), Math 345 (Complex Variables). Students who have not yet taken some of these courses, but have taken and demonstrated strong ability in some 200-level pure mathematics courses such as Math 220 (Discrete Math), Math 225 (Chaos and Fractals), or Math 271 (Linear Algebra) may also be suited for the program. All participants will spend a portion of the summer reading and learning background material together.

Eligibility: The program is open to any full-time Amherst College undergraduate student. The program is a full-time commitment; participants may not be involved in any other summer program, classes, jobs, research opportunities, etc., even if part time.

Application Form: A completed application form is due by email by February 20, 2015. The application form, with instructions, is available at this link . Applicants will have until March 8th, 2015 to accept or decline an offer.

Topic: Modular forms are central objects of study in number theory. Loosely speaking, they are complex-valued functions, which additionally obey certain symmetry properties with respect to a group action. Here’s one example of a modular form:

m(q) := q^(-1/24)(1 + q + 2q^2 + 3q^3 + 5q^4 + 7q^5 + 11q^6 + 15q^7 + 22q^8 + . . . . . )

While interesting in their own right, modular forms are also often studied due to intrinsic combinatorial or algebraic information that they may possess. For example, consider the integer partitions of a positive integer n, the different ways to write n as a non-increasing sum of positive integers (i.e. the partitions of the number n=4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1). It is well known that integer partitions, which a priori are combinatorial in nature, are intimately connected to modular forms (in particular, to the modular form m(q) shown above). Moreover, the special values of modular forms are known to play important roles (i.e. the values of m(q) and other modular forms can be of great algebraic interest when q is appropriately chosen). Participants will spend a portion of the beginning of the summer reading and learning background material on these topics, with the goal of later exploring these types of topics in an original research project with respect to newer, not so well understood, functions of current interest to number theorists.

Here's a link to a related expository paper by Prof. Folsom: A. Folsom, What is...a mock modular form? , Notices of the Amer. Math. Soc. 57 issue 11 (2010), 1441-1443.

Questions? Feel free to email or see Prof. Folsom with any questions about the program or application.

UPDATE:
Student participants: Caleb Ki ’17, Yen Nhi Truong Vu ’17, Bowen Yang ’18.
Results/Paper: A. Folsom, C. Ki ’17, Y. N. Truong Vu ’17, B. Yang ’18. Strange combinatorial quantum modular forms, Journal of Number Theory 170 (2017), 315-346.
Undergraduate Student Paper Session Award MAA-Mathfest 2015, Washington, DC