The CTNT 2020 Summer School will take place June 8 – June 12 (followed by a conference June 12-14). All talks during the summer school will be at the Pharmacy/Biology Building (PBB 129 and 131), and the coffee breaks will be outside of PBB 129. A campus map pointing to PBB can be found here (Google labels the building as the “School of Pharmacy”).

**Note**: This program is open only to students who are currently attending colleges and universities in North America.

## Goals of the Summer School

The organizers of the summer school hope that the students attending this event will learn fundamental ideas in contemporary number theory and have a sense of some directions of current research. For undergraduates, the summer school will expose them to topics not available in a typical college curriculum and we will encourage applications from students at institutions where advanced topics in number theory are not ordinarily taught. The school will provide a chance for participants to meet fellow students, as well as faculty, interested in number theory.

## Expected Background of Students

**Undergraduate Students**: a semester each of elementary number theory and abstract algebra.**Graduate Students**: a year of abstract algebra, and a semester of algebraic number theory.

## Structure of the Summer School

The summer school will take place at the Storrs campus of the University of Connecticut. Activities will be designed at two levels, targeting advanced undergraduate and beginning graduate students. Lectures will be scheduled so that a student could attend almost all lectures if desired, choosing according to their background and interests. The daily schedule in the summer school will be as shown in the following table.

Schedule:

Time |
PBB 129 |
PBB 131 |

8:15 – 9 | Breakfast | |

9 – 9:50 | Mini-course A | Mini-course B |

9:50 – 10:10 | Coffee Break | |

10:10 – 11 | Mini-course C | |

11:10 – 12 | Guest Lecture | |

12 – 2 | Lunch | |

2 – 2:50 | Mini-course D | |

3 – 3:50 | Mini-course E | |

3:50 – 4:10 | Coffee Break | |

4:10 – 5 | Mini-course F | |

5 – 7 | Dinner | |

After 7 | Evening sessions |

## Lecture series

Each day’s events at the summer school is as follows. The videos for the lectures can be found at the UConn Math YouTube Channel.

**Guest Lectures**: Each day will have a plenary talk, where a number theorist will give an overview (accessible to advanced undergraduates and beginning graduate students) of a current trend in number theory. Titles of the lectures and speakers:- June 8: Charlotte Chan (MIT)
**Title**:

**Abstract:**

- June 9: Taylor Dupuy (UVM)
**Title**:

**Abstract**:

- June 10: Jennifer Balakrishnan (BU)
**Title**:

**Abstract**:

- June 11: Jeremy Teitelbaum (UConn)
**Title**:

**Abstract**:

- June 8: Charlotte Chan (MIT)
**Mini-course A**: “*p*-adic functions on Z” by Liang Xiao (Peking University). The course will introduce the ring of_{p}*p*-adic integers Zand then the space of continuous p-adic valued functions on Z_{p}, and its dual space; for these we will introduce the Mahler basis and the Amice transform. The course will finish by constructing the Kubota-Leopoldt_{p}*p*-adic L-function.- Lecture notes.
- Exercises.

**Mini-course B**: “Sieves” by Brandon Alberts (UConn). This course is an introduction to sieving methods in number theory. We will introduce the language and tools used to put sieves on a rigorous footing (asymptotic notation, arithmetic functions, partial summation, etc.) and work through examples of sieves in the lectures. The main examples we will discuss are the Eratosthenes-Legendre sieve and Brun’s combinatorial sieve.- Resources.

- Resources.
**Mini-course C**: “Computations in Number Theory Research” by Álvaro Lozano-Robledo (UConn). This course will serve two purposes. First, we will learn how to use the software packages SageMath and Magma for number-theoretic computations (involving primes, number fields, Galois groups, elliptic curves, curves over finite fields, etc). In addition, the lectures will showcase examples where computations have been an integral part of published research.- Resources.

- Resources.
**Mini-course D**: “Local Fields” by Christelle Vincent (University of Vermont). In this course we will introduce local fields, classify them, and investigate their structure. We will then discuss Henselian fields and Newton polygons, and finally ramification in extensions of local fields. If there is time, we will discuss the decomposition and inertia subgroups of the Galois group of a Galois extension of local fields.- Resources.

- Resources.
**Mini-course E**: “Infinite Galois Theory” by Keith Conrad (UConn). We will describe Galois theory for algebraic field extensions of infinite degree. The Galois correspondence is between intermediate fields and closed subgroups of the Galois group, where “closed” is relative to a topology on the Galois group. The background for this course is finite Galois theory and point-set topology.- Resources.

- Resources.
**Mini-course F**: “Curves over Finite Fields” by Soumya Sankar (Univ. Wisconsin-Madison). This course is an introduction to curves over finite fields and their invariants. We will start with Jacobians of curves and some related structures, such as Tate modules and*p*-divisible groups, and the action of Frobenius on them. Then we will look at tools used to study curves over finite fields, such as Newton polygons and the Hasse-Witt matrix, and how to compute them.- Resources.

- Resources.
**Other sessions**: Participants will have time scheduled outside of the lectures to discuss exercises or review lecture notes from the courses. Instructors and graduate assistants will be available to answer questions. We will also offer the following presentations:- Beamer tutorial: we will cover basic guidelines for creating slide talks using Beamer.
- Graduate school preparation panel: we will give advice and answer questions about the process of applying to graduate school and choosing graduate programs.
- Graduate school advising panel: we will give advice and answer questions about the process of selecting a research area and picking a thesis advisor.