Summer Undergraduate Research in Number Theory at Amherst 2018

Description: Amherst College undergraduate students are invited to apply to spend 7 weeks of summer 2018 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 11 - July 27, 2018. 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.

Funding: Participants will receive a stipend of $440/week for 7 weeks. Participants will also receive (at not cost to them) on-campus housing, subject to availability after formally applying to Amherst College Housing. Included in the summer 2018 housing contract is a partial meal plan (number of meals TBD, at no cost to the student for the included meals). This funding is provided by NSF CAREER Grant DMS-1449679.

Prerequisites: By the start of summer, applicants should have taken, and demonstrated strong ability in, at least two of Math 350 (Groups, Rings and Fields), Math 355 (Analysis), Math 345 (Complex Variables), or Math 310 (Intro. to the Theory of Partitions). Math 460 (Analytic Number Theory), Math 281 (Combinatorics), or Math 250 (Number Theory) may also be useful, but are not required. 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. Current seniors who will graduate in Spring 2018 are not eligible, however. 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 21, 2018. The application form, with instructions, is available at this link . Applicants will have until early March 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). So called "q-series" (infinite power series in the variable q) and their analytic properties are also studied independent of whether or not they are modular forms. 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.

Here are related papers you may wish to take a look at:

(1) An 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.

(2) Results of Summer 2015 research group: 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 .

(3) Results of the Summer 2017 research group: M. Barnett `18, A. Folsom, O. Ukogu `18, W. Wesley `18, H. Xu `18, Quantum Jacobi forms and balanced unimodal sequences, Journal of Number Theory 186 (2018), 16-34.

(4) Results of the SUMRY 2014 research group: A. Folsom, Y. Homma, J. Ryu, and B. Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 iss. 5 (2016), 1482-1506.

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

UPDATE:
Student participants: Greg Carroll ’20, James Corbett ’19, Ellie Thieu ’19.
Results/Paper: G. Carroll ’20, J. Corbett ’19, A. Folsom, and E. Thieu ’19. Universal mock theta functions as quantum Jacobi forms, Research in the Mathematical Sciences, 6:6 (2019), 15pp.