Mathematics Research Communities, American Mathematical Society

3 572Program ID: MRC-HYPERBOLIC [#572]
Program Title: Number Theoretic Methods in Hyperbolic Geometry Week 2b, June 10-16, 2018
Program Location: Rhode Island, United States [map]
Application Deadline: 2018/02/15** (posted 2017/07/07, updated 2017/05/12, listed until 2018/10/31)
Program Description:    

About the Mathematics Research Communities:

Mathematics Research Communities (MRC), a program of the American Mathematical Society (AMS), nurtures early-career mathematicians--those who are close to finishing their doctorates or have recently finished--and provides them with opportunities to build social and collaborative networks through which they can inspire and sustain each other in their work.

The structured program is designed to engage and guide all participants as they start their careers. For each topic, the program includes a one-week summer conference, a Special Session at the next Joint Mathematics Meetings, and a longitudinal study of early career mathematicians.

The summer conferences of the MRC are held at the Whispering Pines Conference Center in West Greenwich, Rhode Island, where participants can enjoy the natural beauty and a collegial atmosphere. Those accepted into this program will receive support (full room and board at Whispering Pines Conference Center and airfare or partial airfare) for the summer conference, and will be partially supported for their participation in the Joint Mathematics Meetings which follow in January 2019.

ELIGIBILITY: Individuals within one to two years prior to the receipt of their PhDs, or up to five years after receipt of their PhDs, are welcome to apply.  Most of those supported by NSF funds to participate in the MRC program will be US-based, that is, employed by or a full-time student at a US institution at the time of the MRC summer conference. However, the terms of the grant allow for a limited number of individuals who are not US-based. A few international participants may be accepted. Women and underrepresented minorities are especially encouraged to apply.  All participants are expected to be active in the full MRC program.

For any program, fellowship, prize or award that has a maximum period of eligibility after receipt of the doctoral degree, the selection committee may use discretion in making exceptions to the limit on eligibility for candidates whose careers have been interrupted for reasons such as family or health. Therefore, an applicant who has had to slow down or temporarily stop his or her career for personal reasons may request to be considered for an extension in the amount of time after the PhD degree. Please send exception requests to

  • Completed on-line application form
  • One (1) reference letter submitted by a professor or supervisor who knows the applicant and can address how the applicant will benefit from, and contribute to, the MRC program.

Note that all applicants will be notified of their status by May 1, 2018.

Week 2: June 10 – 16, 2018-- Number Theoretic Methods in Hyperbolic Geometry

Benjamin Linowitz, Oberlin College
David Ben McReynolds, Purdue University
Matthew Stover, Temple University

The canonical example of an arithmetic lattice is the modular group PSL(2, Z), whose deep connections with geometry and number theory (among many other areas) have been of profound interest for well over a century. Geometric invariants of the modular surface—the quotient of the complex upper half-plane by PSL(2, Z)—are typically paired with objects of equally deep interest in number theory. For example, its volume in its metric of constant curvature ¬-1 is naturally related to a special value of the Riemann zeta function, and the lengths of its closed geodesics are intimately related to class numbers and regulators of real quadratic fields. More generally, rigidity phenomena (Weil, Mostow, etc.) imply that similar connections exist between number theory and the geometry of higher-dimensional hyperbolic manifolds.

The primary focus during the workshop will be to introduce the participants to problems at the interface of geometry and number theory that are currently attracting significant interest, and provide them with the tools necessary to make progress on some open questions. General areas to be discussed include the Laplace eigenvalue spectra and geodesic length spectra of hyperbolic 2- and 3-manifolds, growth of the systole of a hyperbolic manifold, and the ‘realization problem’ for trace fields of hyperbolic 3-manifolds. The number theoretic techniques that we will use to address these problems make use of quaternion algebras over number fields, Mahler measures of algebraic integers, and classical results from multiplicative number theory. The ultimate goal of this workshop is to start a dialogue between young mathematicians from different backgrounds that will lead to new and long-lasting collaborations between fields that have a great deal to say to one another. No background in either subject is expected.

Application Materials Required:
Submit the following item online at this website to complete your application:
And anything else requested in the program description.

Further Info:
800-321-4267 x 4113
Electronic submission of reference letters is requested.
If this is not possible, contact

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