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.
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.
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.
AWS 2025 will be held March 8-12, 2025 at the University of Arizona in Tucson, AZ.
Registration will open mid-September and will close on November 15th.All participants must abide by the University of Arizona nondiscrimination and anti-harassment policy.
Please refer to University of Arizona academic calendars for the tentative dates of future Winter Schools. (The AWS begins on the first day of the University of Arizona's Spring Recess in any given year.)