Preliminary Arizona Winter School 2024: Symmetries of root systems and local fields

The Preliminary Arizona Winter School (PAWS) is a virtual program on topics related to the upcoming AWS, with an intended audience of advanced undergraduate students and junior graduate students.

Registration: You can apply for the program here. The deadline to apply is July 15th, 2024.

September 23rd — November 22nd, 2024

PAWS 2024 consists of two concurrent lecture series.

• SHORTTITLE  Melissa Emory: Symmetries of root systems

A root system is a "very symmetrical" set of vectors in n-dimensional Euclidean space. The classical motivation for studying root systems is their role in the classification of semi-simple Lie algebras, i.e. classical and exceptional groups, over the complex numbers. The root datum has connections to the Langlands dual group as well as L-functions. Thus, root systems have connections to representation theory, number theory, algebra, geometry, and physics.

We will outline the classification of root systems and discuss Dynkin diagrams. We will also define the Weyl group of a root system and show some examples of using root systems in the Langlands program.

• SHORTTITLE  Catherine Hsu: Local fields

In 1897, Kurt Hensel introduced the p-adic numbers as a way to apply techniques involving power series within the context of number theory. The p-adic numbers are an example of a local field---that is, a field arising as a suitable completion of either a number field or a function field over a finite field. In modern number theory, many deep questions about the arithmetic of number fields have been approached by first investigating their local versions, with applications ranging from Diophantine geometry to the Langlands program.

Starting from discrete valuations, this course will explore the theory of local fields. The first half of the course will focus on arithmetic properties of local fields, highlighting applications of Hensel's lemma to quadratic forms. In the second half of the course, we will shift to the Galois theory of local fields, with an emphasis on ramification groups. Time permitting, we will conclude the course with an application of local Galois theory to the proof of the Kronecker--Weber Theorem.

Format: The program will run for 9 weeks, alternating between 5 weeks of lectures and 4 weeks of problem sets (with a TA session)

Each lecturer will be accompanied by graduate student assistants, who will be in charge of writing weekly problem sets and facilitating weekly, hour-long problem solving and discussion meetings with groups of students. Recommended background for the program is a first course in abstract algebra.

The school will feature an online (Zulip) discussion board where students can ask questions and interact with the speakers and assistants outside of scheduled meeting times.

We will facilitate additional virtual events, some purely social to build a community, and some more structured sessions on timely and pertinent topics like "What is graduate school in Math like?", "Tips for applying to graduate school this Fall," "How do I navigate the first year of graduate school?" "How do I look for an thesis advisor?", "How to get the most out of the upcoming AWS."

We encourage undergraduate students to take their PAWS course as an independent study with a faculty member at their home institution.

• Organizers: Renee Bell, Isabel Vogt, David Zureick-Brown (main organizers) with Alina Bucur, Brandon Levin, Padma Srinivasan, Anthony Várilly-Alvarado, and Hang Xue
• Funded by the National Science Foundation.