Summer School

The CTNT 2026 Summer School will take place June 1 – June 7. 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”).

  • Registration page.

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/Problem sessions
    12 – 2 Lunch
    2 – 2:50 Mini-course C
    3 – 3:45 Problem sessions
    3:50 – 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 previous lectures can be found at this 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 2: Griff Elder (UConn) on “TBA”
        • June 4: Christelle Vincent (UVM) will speak on “TBA”
    • Mini-course A (Analytic Number Theory): “Analytic Aspects of Quadratic Forms” by Keith Conrad (UConn). Quadratic forms have been part of number theory since its beginning. This course will discuss some of the ways that quadratic forms are studied using analytic techniques like theta-functions and zeta-functions, with applications to topics such as representation numbers and class numbers.
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    • Mini-course B (Algebraic Number Theory): “The Kronecker–Weber Theorem and Abelian Extensions” by Lori Watson (Trinity College). In this course we will introduce students to the study of number fields and their extensions and, in particular, their abelian extensions. We will discuss the Kronecker–Weber theorem and will explore its possible generalizations to other fields, such as imaginary quadratic fields.
    • Mini-course C (Arithmetic Geometry): “Elliptic Curve Cryptography” by Alvaro Lozano-Robledo (UConn). In this course we will start by reviewing some elementary cryptographic methods that use number theory, such as RSA and the classical Diffie–Hellman key exchange, and then we will give a quick introduction to elliptic curves (over finite fields) with the goal of describing more modern algorithms, such as the elliptic curve Diffie—Hellman key exchange, and isogeny-based cryptographic methods. Theoretical and computational exercises will be suggested.
    • Mini-course D (Computational Number Theory): “Magma and the LMFDB” by
      Eran Assaf (MIT). This will be an overview of the Magma Computational Algebra System and the L-functions and Modular Forms Database (LMFDB), discussing their application to mathematical research in algebra, number theory and arithmetic geometry. Students will learn how to query the LMFDB to obtain answers to non-trivial questions, and evidence for well-known conjectures. The mini-course will also use Magma to compute class groups of rings, fields and quaternion algebras, the torsion and rank of elliptic curves, as well as rational points on genus 2 curves, and interesting spaces of modular forms.

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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.