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## UNCG Summer School in Computational Number Theory 2018

### Algorithms for Extensions of Large Degree

From May 28 to June 1, 2018, the University of North Carolina at
Greensboro will host the UNCG Summer School in Computational Number
Theory: **Algorithms for Extensions of Large Degree**.

The summer school in computational number theory fills a gap in the education of many graduate students. Most graduate courses in number theory take a mainly theoretic approach with very little emphasis on the computational aspects of the subject. The goal of the UNCG Summer School in Computational Number Theory is to complement this with a constructive-algorithmic approach. Many of the algorithms used for number theoretic computations are non-trivial, which makes it difficult to cover them in a standard course.

Algorithms for extensions of global and local fields are the backbone of computational number theory. Improvements in computing power have made it feasible to conduct computations in larger and larger degree. As the complexity of algorithms for field extensions depends on the degree of the extensions, this has increased the interest in asymptotically fast algorithms. Among others we will consider algorithms for integral bases computation and ideal arithmetic.

On a typical day, external and local experts give talks in the morning, and in the afternoon students solve problems related to this material. The talks early in the week introduce the students to the subject. Talks later in the week cover 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 are aimed at increasing the studentsâ€™ understanding of the material by working with it.

#### Application

We expect to be able to provide support for approximately ten graduate students (travel, dorm accomodation, food).
Interested students should complete the online
application
and have a letter of reference sent via email to batanged@uncg.edu by
**March 22, 2018**. We will respond with funding decisions by April 13, 2018.

#### Speakers

- Peter Bruin (Universiteit Leiden)
- Claus Fieker (Technische Universität Kaiserslautern)
- Jordi Guardia (Universitat Politècnica de Catalunya)

#### Software

- +Ideals, a package for ideal arithmetic in number fields for Magma
- Hecke, a software package for algebraic number theory maintained by Claus Fieker and Tommy Hofmann written in the Julia programming language and based on the computer algebra package Nemo.
- modgalrep , a package for computing in Jacobians of projective curves over finite fields, with applications to modular Galois representations, based on PARI/GP

#### Draft of Schedule

All talks will take place in room 213 in the Petty Building (campus maps and directions). The problem sessions will take place in Petty 213 and the computer lab in Petty 211.

Time | Sunday 5/27 | Monday 5/28 | Tuesday 5/29 | Wednesday 5/30 | Thursday 5/31 | Friday 6/1 |
---|---|---|---|---|---|---|

9:00 | Arrival | Welcome | Coffee | |||

9:30 | Participants: Introductions | Peter Bruin | Jordi Guàrdia | Claus Fieker | Peter Bruin | |

10:45 | Coffee | |||||

11:15 | Claus Fieker | Claus Fieker | Peter Bruin | Jordi Guàrdia | Final Reports | |

12:30 | Lunch | |||||

14:00 | Jordi Guàrdia | Problem Session | Excursion | Problem Session | ||

15:30 | Problem Session | |||||

17:00 | Student Reports | Student Reports | ||||

Old Town Draught House | Hillbilly Hideaway |

#### Organizers

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

#### Acknowledgements

The summer school in computational number theory is supported by UNCG and the NSF (DMS-1602025).