Summer School

The CTNT 2024 Summer School will take place June 10 – June 16. 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 can attend 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 131
    8:15 – 9 Breakfast
    9 – 9:50 Mini-course A
    9:50 – 10:10 Coffee Break
    10:10 – 11 Mini-course B
    11:10 – 12 Guest Lecture
    12 – 2 Lunch
    2 – 2:50 Mini-course C
    3 – 3:30 Mini-course E
    3:30 – 4:00 Break
    4:00 – 4:50 Mini-course D
    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 this YouTube Channel. (Note: the mini-course on Adeles and Ideles was delivered on the board, and it was not recorded.)

    • 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 11: Jeremy Teitelbaum (UConn) will speak on “Factoring with elliptic curves.”
        • June 13: David Pollack (Wesleyan University) will speak on “Dirichlet’s theorem on primes in arithmetic progressions.”
    • Mini-course A: “Using Quadratic Reciprocity” by Keith Conrad (UConn). In this course, we will describe some ways in which quadratic reciprocity can be applied to solve problems in number theory and related areas of mathematics. We will also see how ideas that were introduced to prove quadratic reciprocity have been influential in the development of number theory.  
    • Mini-course B: “Adeles and Ideles” by Lori Watson (Trinity College). In this course we will introduce the ring of the adeles and the idele group for the rational numbers Q. We will discuss the field of the p-adic numbers and its subring of the p-adic integers. We will then discuss the construction of the adeles and ideles and some of their uses.
    • Mini-course C: “Class Field Theory” by Christelle Vincent (University of Vermont). We will begin with a discussion of local class field theory, explaining the theory in some detail for concreteness. Then we will formalize what we have seen into an “abstract class field theory,” following Neukirch, and finally very briefly gesture at the global theory for number fields.
    • Mini-course D: “Introduction to Elliptic Curves” by Alvaro Lozano-Robledo (UConn). This will be an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves.
    • Mini-course E: “Computations in Number Theory.” 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.
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    • 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.