Summer School 20-Online

The CTNT 2020 Summer School will take place June 8 – June 12 (followed by a conference June 12-14). Due to the current situation, CTNT 2020 will be an entirely online program.

REGISTRATION FOR THE SUMMER SCHOOL IN NOW CLOSED. Students who applied for the in-person CTNT 2020 summer school have been notified by email about the registration process for the online summer school.

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

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 (tentative — it might be modified in the next few weeks before the program) Monday thru Thursday — all times are US/Eastern (UTC-4):

    9:30 – 10 Informal chat
    10 – 10:50 Mini-course A p-adic functions on Zp” by Liang Xiao
    11:10 – 12:00 Mini-course B “Computations in Number Theory Research” by Álvaro Lozano-Robledo
    12 – 1:30 Lunch

    (followed by social video call)

    1:30 – 2:20 Mini-course C “Sieves” by Brandon Alberts
    2:40 – 3:30 Mini-course D “Curves over Finite Fields” by Soumya Sankar
    3:30 – 4:00 Coffee Break
    4:00 – 4:50 Mini-course E “Infinite Galois Theory” by Keith Conrad
    5 – 7 Dinner

    (followed by social video call, and working groups)


    Lecture series

    Each mini-course will have 4 lectures (from Monday to Thursday). The videos for the lectures can be found at the CTNT 2020 YouTube playlist for mini courses.

    • Mini-course A: “p-adic functions on Zp” by Liang Xiao (Peking University). The course will introduce the ring of p-adic integers Zp and then the space of continuous p-adic valued functions on Zp, 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-adic L-function.
    • Mini-course B: “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.
    • Mini-course C: “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.
    • Mini-course D: “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.
      • Course notes (will be available at a later time — participants can find them in the Piazza page for the course)
      • Videos: YouTube playlist.
    • 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.
    • Other sessions: Participants will have time scheduled outside of the lectures to discuss exercises or review lecture notes from the courses, via video calls. Instructors and graduate assistants will be available to answer questions. We will also offer the following presentations:
      • Graduate school preparation panel: we will give advice and answer questions about the process of applying to graduate school and choosing graduate programs. We will also give advice and answer questions about the process of selecting a research area and picking a thesis advisor.