2019 Oregon Number Theory Days- Winter Meeting

The winter meeting of Oregon Number Theory Days will take place on Saturday, February 16 at Oregon State University, in Corvallis, Oregon. This is a triannual seminar rotating between Oregon State University, Portland State University, and the University of Oregon. 

Program:

11:00-12:00 -- John Voight (Dartmouth University), Lecture I: Definite quaternion orders with stable cancellation

12:00-1:00 -- Lunch

1:00-2:00 -- John Voight (Dartmouth University), Lecture II: Computing Belyi maps

2:15-2:45 -- Piper H (University of Hawaii at Manoa), Lecture I: Equidistribution of shapes of (certain) number fields

2:45-3:30 -- Graduate student poster session and coffee break

3:30-4:15 -- Piper H (University of Hawaii at Manoa), Lecture II: How to Become a Liberated Mathematician in 13 Painful Years

Abstracts: 

• John Voight, Lecture I: Definite quaternion orders with stable cancellation Gauss conjectured (in the language of binary quadratic forms) that there are finitely many imaginary quadratic orders of class number 1. There are countless variants of this problem, involving mathematics that is both deep and ongoing. We will survey versions of the class number problem for quaternion orders. In particular, we enumerate all orders with cancellation in the stably free class group. This is joint work with Daniel Smertnig.

• John Voight, Lecture II: Computing Belyi maps A Belyi map is a finite, branched cover of the complex projective line that is unramified away from 0, 1, and infinity. Belyi maps arise in many areas of mathematics, and their applications are just as numerous. They gained prominence in Grothendieck's program of dessins d'enfants, a topological/combinatorial way to study the absolute Galois group of the rational numbers. In this talk, we survey computational methods for Belyi maps, and we exhibit a uniform, numerical method that works explicitly with power series expansions of modular forms on finite index subgroups of Fuchsian triangle groups. We also present a beta version of a database of Belyi maps (http://beta.lmfdb.org/Belyi). This is joint work with Michael Klug, Michael Musty, Sam Schiavone, and Jeroen Sijsling.

• Piper H, Lecture I: Equidistribution of shapes of (certain) number fields In her talk, Piper will introduce the ideas that there are number fields, that number fields have shapes, and that these shapes are everywhere you want them to be. This result is joint work with Manjul Bhargava and uses his counting methods which currently we only have for cubic, quartic, and quintic fields. She will sketch the proof of this result and leave the rest as an exercise for the audience. (Check your work by downloading her thesis!).

• Piper H, Lecture II: How to Become a Liberated Mathematician in 13 Painful Years Piper never wanted to be liberated. She would have much preferred to be conventionally successful, living by other people's standards. Though she tried, she couldn't make herself fit. You could say she has some complaints. Now she has a story to tell. A story of failure and how sometimes failure is the same as leadership.