Summer Undergraduate Research in Number Theory at Amherst 2020

Description: Amherst College undergraduate students are invited to apply to spend 8 weeks of summer 2020 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. It is anticipated that three students will participate.

Dates: June 8 - July 31, 2020. 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 the national conference MAA Mathfest July 29 - August 1, 2020, in Philadelphia, PA. (Funding will be provided.)

Note. Due to Covid-19, and with permission of NSF, this program will run virtually during the 8 week program. MAA Mathfest was canceled due to Covid-19.

Funding: Participants will receive a stipend of $540/week for 8 weeks, for a total of $4320. 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 2020 housing contract is a partial meal plan, expected to be 14 meals per week (lunch and dinner every day), at no cost to the student for the included meals. This funding is provided by Prof. Folsom's NSF Grant DMS-1901791.

Note. Due to Covid-19 and the resulting virtual nature of this program, housing/meals not provided.

Prerequisites: Preference given to applicants who, by the start of summer, 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 equivalent. Math 460 (Analytic Number Theory), Math 310 (Intro. to the Theory of Partitions), 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 2020 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 15, 2020. 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 some related papers (authored by Prof. Folsom and past students) for more:

(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) Another (partially expository) paper by Prof. Folsom: A. Folsom, Asymptotics and Ramanujan’s mock theta functions: then and now, Philosophical Transactions of the Royal Society A, accepted for publication (2019). 13pp.

(3) Results of the Summer 2018 research group: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.

(4) 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.

(5) 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 .

(6) 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: Elizabeth Pratt '22, Noah Solomon '22, Andrew Tawfeek '21E.
Results/Paper: A. Folsom, E. Pratt '22, N. Solomon '22, A.R. Tawfeek '21E, Quantum Jacobi form and sums of tails identities, Research in Number Theory, 8:8 (2022). 24pp.
MAA Outstanding Undergraduate Student Poster Session Award, Joint Mathematics Meetings 2021(Poster 22)