Summer School 22

The CTNT 2022 Summer School will take place June 6 – June 12. 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 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 6: Keith Conrad (UConn), “Wieferich primes”
        • June 7: Anna Medvedovsky (Boston University), “Zero sets of linear recurrence sequences”
        • June 8: Jeremy Teitelbaum (UConn), “p-adic Fourier Theory and applications”
    • Mini-course A: “Algebraic Number Theory” by Hanson Smith (UConn). This will be an introductory crash course on algebraic number theory. We will cover the basic definitions of number fields and number rings with ample examples. The mini-course will culminate with some of the foundational ideas on prime splitting and Galois theory.  
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    • Mini-course B: “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.
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    • Mini-course C: “100 Years of Chebotarev Density” by Keith Conrad (UConn). The Chebotarev density theorem, which was proved in Chebotarev’s PhD thesis in 1922, is an important result connecting algebraic and analytic number theory. It is a “non-abelian” generalization of Dirichlet’s theorem on primes in arithmetic progressions.  In the course we will explain what the theorem says, see different approaches to proving it, and describe some of its many applications.
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    • Mini-course D: “An Introduction to Galois Representations” by Alvaro Lozano-Robledo (UConn). We will introduce the basic properties of the absolute Galois group of Q, and its representations (which are called Galois representations). Then, we will produce examples of Galois representations, concentrating on those coming from elliptic curves. In particular, we will discuss a recent classification of ell-adic Galois representations attached to elliptic curves by Rouse, Sutherland, and Zureick-Brown.
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    • Mini-course E: “Computations in Number Theory.”
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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.