Preliminary Arizona Winter School 2022: Heights and Model Theory
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.
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, Hang Xue, with Alina Bucur, Brandon Levin, Anthony Várilly-Alvarado, and David Zureick-Brown.
- Funded by the National Science Foundation.
October 3rd — November 11th, 2022
PAWS 2022 consists of two concurrent six-week lecture series.-
Ronnie Nagloo: Introduction to model theory with applications
- Video:
- Notes:
- Problem Sets:
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Padmavathi Srinivasan: Heights in Diophantine geometry
- Video:
- Lecture 1: stream, download
- Lecture 2: stream, download
- Lecture 3: stream part 1, stream part 2, download
- Lecture 4: stream, download
- Lecture 5: stream part 1, stream part 2, download
- Lecture 6: stream, download
- Notes:
- Problem Sets:
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Assisted by
Sebastian Eterović, Elliot Kaplan, Kevin Zhou (Nagloo), and Niven Achenjang, Juanita Duque Rosero, Carlos Rivera, Marley Young (Srinivasan)
Model theory is a branch of mathematical logic dealing with abstract structures, historically with connections to other areas of mathematics. The developments, over the past several decades, have allowed for a strengthening of those connections as well as new striking applications to areas such as diophantine and analytic geometry, algebraic differential equations, and combinatorics. This course will serve as an introduction to the basics of model theory, with a view towards some of the above applications.
The height of a rational number is a measure of its arithmetic complexity. For example, although the numbers 5 and 500000/100001 are close, the second is arithmetically more complex, and has a much larger height. The height of a rational number is easy to define -- it is simply the maximum of the absolute value of the numerator and the denominator when the number is expressed in lowest form. It is not immediately clear how one can extend this definition to more general algebraic numbers such as the squareroot of 2. In this series of lectures, we will develop the theory of heights of algebraic numbers, and present "Weil's height machine" for defining heights more generally for solutions to systems of polynomial
equations in algebraic numbers (i.e., heights of algebraic points on varieties).
Height functions are a key tool in proving many important finiteness theorems in Diophantine Geometry. The main property of height functions is that there are only finitely many points of bounded height and degree on any given variety. Understanding how quickly the number of points grow as the height grows for various classes of varieties is an active area of research in number theory today! As an application of the theory of heights, we will prove the Mordell--Weil theorem for elliptic curves, namely that the set of rational solutions to cubic equations such as y^2 = x^3 - 2x + 2 is finitely generated.