## UNCG Summer School in Computational Number Theory 2014

### Modular Forms and Geometry

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 studentsâ€™ understanding of the material by working with it.

#### Schedule

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 (exercises) | 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 (exercises) | 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. The PANTS 13 pages contain a list of local restaurants.

#### Participants

- William Cocke (BYU)
- Thomas Alden Gassert (UMass)
- Jeffery Hein (Dartmouth College)
- Brian Hwang (Cal Tech)
- Andrew Jones (Sheffield)
- Tianyi Mao (CUNY)
- James Martin (North Texas)
- Jolanta Marzec (Bristol)
- Richard Moy (Nortwestern)
- Jesse Patsolic (Wake Forest)
- James Ricci (Wesleyan)
- Karen Taylor (Bronx Community College)
- Ka Lun (Allan) Wong (Hawaii)

##### Locals

- Paul Duvall (UNCG)
- Lance Everhart (UNCG)
- Ricky Farr (UNCG)
- Paula Hamby (UNCG)
- Jonathan Milstead (UNCG)
- Benjamin Manifold (UNCG)
- Brian Sinclair (UNCG)

##### Speakers

- Avner Ash (Boston College)
- Paul Gunnells (UMass)
- Matt Greenberg (Calgary)

##### Organizers

- Brett Tangedal (UNCG)
- Dan Yasaki (UNCG)
- Filip Saidak (UNCG)
- Sebastian Pauli (UNCG)

#### Acknowledgements

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