Analytic Combinatorics,  Modular  Forms,   and  Number  Theory  REU

June 23 - August 1, 2025.

The 2025 USC Number Theory REU is a six-week research experience for undergraduates, held in Columbia, SC (Cola City). While federal funding will not proceed this summer, thanks to generous support from Jane Street, we are able to offer a scaled-down version of the program. Participants will work in two groups, exploring topics in analytic combinatorics, modular forms, and number theory. These include problems in combinatorial and number-theoretic sequences, modular form congruences, integer partitions, and related research areas.

As a participant, you will join the research program of a faculty mentor, working alongside graduate students and engaging in regular discussions with your mentor. The program is designed to provide a collaborative and immersive research experience, fostering close mentorship and meaningful academic interactions.

Each summer, 2-3 plenary speakers will be invited to share insights from their research and their experiences with REU programs. The program also features a weekly professional development seminar series, enrichment activities, and social events with other REU students. At the end of the summer, participants will present their research at the Summer Research Symposium, joining students from other summer research programs at USC.

The program will run from June 23 to August 1, 2025, and will provide a small stipend to participants.

Participation & Program Details

  • Dates:   Monday, June 23 to Friday, August 1, 2025.

  • Location:   Mathematics Department, University of South Carolina, Columbia, SC.

  • Mentors:

      Matthew Boylan     University of South Carolina  
      Wei-Lun Tsai     University of South Carolina  

  • Plenary Speakers:

    Hui Xue
    (July 7)     		Clemson University Talk 1: Zeros of modular forms
    We will investigate the locations and interlacing properties of certain modular forms, such as Eisenstein series and their simple combinations. A prototype result in this direction is the Rankin and Swinnerton-Dyer's elegant proof that all zeros of an Eisenstein series lie on the lower arc of the fundamental domain.

    Talk 2: Zeros of period polynomials
    We will study the locations and interlacing properties of period polynomials associated to Hecke eigenforms. These polynomials satisfy a symmetrical functional equation similar to that of the Riemann zeta function. Their zeros are also located on the axis of symmetry, as predicted by the Riemann Hypothesis.

    Robert Dicks
    (July 15)     		Clemson University Talk 1: Congruences like Atkin's for the partition function: preliminaries
    In this talk, the author gives relevant background material for understanding the recent results on the partition function by Ahlgren-Allen-Tang in 2021. This includes standard facts from the theory of modular forms, Galois theory, and Galois representations in characteristic 0 and positive characteristic.

    Talk 2: Congruences like Atkin's for the partition function: generalizations
    In this talk, we complete a sketch of the proof of the main result from the work of Ahlgren-Allen-Tang. We then discuss recent work of the author establishing a wide generalization of these congruences using the Shimura lift for the eta multiplier he developed jointly with Nickolas Andersen and Scott Ahlgren.

    Frank Thorne
    (July 22)     		University of South Carolina Talk 1: The magic of integral binary quadratic forms
    Mathematicians like Lagrange, Legendre, and Gauss spent a great deal of time studying the arithmetic of integral binary quadratic forms. Such a simple structure turns out to hold lots of secrets! I will give an overview of what they and others discovered.

    Talk 2: An introduction to arithmetic statistics
    "Arithmetic statistics" is a fascinating branch of number theory, and these days brilliant papers are being posted to the arXiv faster than anyone can read them all. I will give an overview of what all the fuss is about, and describe a few contemporary research directions.

  • Graduate Student Mentors:

      Swati     University of South Carolina  
      Tapas Bhowmik     University of South Carolina  

  • Application and Deadline:

      The online application for summer 2025 is available now, and applications are due April 11, 2025, at 11:59 PM EST. To apply, please complete the following:

    • Additional Resources

  • Directions to campus:   The University of South Carolina is locatedin Columbia, South Carolina. The mathematics department is in LeConte College, at the corner of Greene and Pickens Streets.You can also use (for example) Mapquest for directions.

  • Parking:   Visitor meter lots (in green) are free on weekends. Summer parking permits can be purchased here.

  • Research Workshops:   Summer research seminar series schedule is here.

  • Research Symposium:   Summer research symposium is here.

  • Organizers:   Matthew Boylan • Wei-Lun Tsai.

    Please feel free to e-mail us (uscntreu@gmail.com) if you have any questions.

    Research Highlights

    Three papers based on student research projects are being finalized and will be posted here soon.

    This REU program is made possible through the generous support of the following sponsors. Thank you!

    Last updated: 8/1/2025.

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