Research: Number Theory

Summer School 2014: Geometry and Modular Forms

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From May 19 to May 23, 2014, the University of North Carolina at Greensboro is hosting a summer school entitled Modular Forms and Geometry.

Modular forms play an increasingly important role in number theory and arithmetic geometry. The main focus of the summer school is the study of computional aspects of modular forms and related objects. Topics may include classical modular forms and modular symbols, group cohomology and Galois representations, and lattice enumeration and isometry testing techniques to compute with spaces of modular forms for compact forms of classical groups.

On a typical day, external and local experts gave talks in the morning, and in the afternoon students solved problems related to this material. The talks early in the week introduced the students to the subject. Talks later in the week covered related areas of current research and unsolved problems. The problems given to the students might be of a theoretical nature but could also involve programming problems and computer experiments. All problems were aimed at increasing the students’ understanding of the material by working with it.


All talks took place in Room 219 in the Petty Building (campus maps and directions). Directions from the Biltmore Hotel to Petty. (Switch to walking directions.)

Time Monday 5/19 Tuesday 5/20 Wednesday 5/21 Thursday 5/22 Friday 5/23
9:00 Welcome Coffee Coffee Coffee Coffee
9:30 Participants: Introductions slides Paul Gunnells: Classical modular symbols II Matt Greenberg: Algebraic modular forms
Paul Gunnells: Classical modular symbols III Matt Greenberg: Lattice neighbors and Hecke operators
10:45 Coffee Coffee Coffee Coffee Coffee
11:15 Paul Gunnells: Classical modular symbols I
Avner Ash: Galois representations, group homology, and reciprocity (exercises) Avner Ash: How to compute Hecke operators for GL(n)/Q Avner Ash: The case of reducible Galois representations Rump session
12:30 Lunch Lunch Lunch Lunch Lunch
14:00 Matt Greenberg: Spherical Harmonics and Hecke operators (exercises) Problem Session Excursion to Battleground park Problem Session Problem Session
15:00 Problem Session Problem Session Problem Session Problem Session
19:00 ACC Baseball

Lecture notes

  • Ash lectures: Notes by Dan Yasaki from three lectures by Avner Ash.
  • Gunnells lectures: Expanded notes from three lectures given by Paul E. Gunnells.

Local information

Information for visitors, including directions to UNCG and Petty Building.

Participants & Faculty






NSF LogoNational Security Agency LogoThe summer school in computational number theory is supported by UNCG, the NSA (H98230-13-1-0253), and the NSF (DMS-1303565).