Summer Undergraduate Research in Number Theory at Amherst 2021

Description: Amherst College undergraduate students are invited to apply to spend 8 weeks of summer 2021 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 or four students will participate.

Dates and Format: June 14 - August 6, 2021. While this would ordinarily be an on-campus residential program, due to Covid-19 and with permission of NSF, it is expected that this program will run virtually during the 8 week summer 2021 program.
The program will likely include attending and/or presenting at a virtual or in-person mathematics conference (to be determined this summer as conferences are announced).
Participating students will need a suitable daily workspace, a computer with internet access and webcam for zoom meetings, and a tablet for virtual collaboration (which will be provided to participants who need one on loan). If any of these things pose a challenge for you, please let Prof. Folsom know in your application form (you may still apply).

Funding: Participants will receive a stipend of $570/week for 8 weeks (paid bi-weekly), for a total of $4560. Funding is provided by Prof. Folsom's NSF Grant DMS-1901791.

Prerequisites: Preference given to applicants who, by the start of summer, have taken and demonstrated strong ability in two or more of Math 350 (Groups, Rings and Fields), Math 355 (Analysis), Math 345 (Complex Variables), or equivalent (though you may apply without). 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. No prior research experience is expected. 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. Current seniors who will graduate in Spring 2021 (or who graduated in December 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 Process: Completed applications are due by Sunday, February 21, 2021, 11:59pm EDT. A complete application consists of two parts:

1. Complete the application form at this link.
2. You must also apply to this position through Amherst's Workday system. Log in to Workday at this link, search for available student positions, and apply to this one (called JR344 Mathematics Summer Student Research Assistants). Note that the application in Amherst's workday system is a formal one, used by Amherst College, and the application there is separate and different from the one in part (1) above. In the Workday application, you should write "N/A" to their question asking you to list references for your position. But, in the application google form in part (1) above you should still answer Prof. Folsom's question there asking you to name one potential reference.

Applicants will have until early March to accept or decline an offer (exact date TBD).

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 is a link to an expository talk Prof. Folsom gave on this topic, plus some related papers (authored by Prof. Folsom and past students) for more:

(1) Recording of Prof. Folsom's MAA Invited Address, 2019 Joint Mathematics Meetings, Baltimore:
A. Folsom, Symmetry, Almost, January 14, 2019, Baltimore Convention Center

(2) 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.

(3) 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.

(4) Results of the Summer 2020 research group:A. Folsom, E. Pratt '22, N. Solomon '22, and A.R. Tawfeek ’21E, Quantum Jacobi forms and sums of tails identities, submitted for publication (2020), 25pp.

(5) 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.

(6) 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.

(7) 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 .

(8) 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: Anna Dietrich ’22, Keane Ng ’23, Chloe Stewart ’22, Shixiong Xu ‘23.
Results/Paper: A. Dietrich '22, A. Folsom, K. Ng '23, C. Stewart '22, S. Xu '23, Overpartition ranks and quantum modular forms, Research in Number Theory 8:45 (2022). 16pp.
MAA Outstanding Student Paper Session Award, MAA-Mathfest 2021