Date: Saturday, February 16, 2019

Location: Oregon State University, Kearney Hall room 112

Program:

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

• 12:00-1:00 -- Lunch

• 1:00-2:00 -- John Voight (Dartmouth College), 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.

If you plan to attend this meeting, please register by filling out this webform.

A very limited amount of funding may be available for regional graduate students. To apply for funding to attend the next meeting, please fill out the above registration webform.

The meeting will be held in Kearney Hall room 112, located near the corner of Monroe Ave. and 14th St. on the campus of Oregon State University. While on-campus parking lots require a permit from Monday through Friday,

Spring 2018 (Portland State University):

• Rachel Pries (Colorado State University) Lecture I, Newton polygons of cyclic covers of the projective line. (video of this lecture)

• Rachel Pries (Colorado State University) Lecture II, Generalizing a Galois action on the homology of the Fermat curve. (video of this lecture)

• Özlem Ejder (Colorado State University): Sporadic points on X_1(n). (video of this lecture)

Winter 2018 (Oregon State University):

• Ken Ono (Emory University) Lecture I, Polya's Program for the Riemann Hypothesis and Related Problems (lecture slides)

• Ken Ono (Emory University) Lecture II, Can you feel the Moonshine? (lecture slides)

• Asif Zaman (Stanford University), A new formulation of the Chebotarev density theorem

• Posters presented by Jetjaroen Klangwang (Oregon State University), Peter Cho-Ho Lam (Simon Frasier University), and Daniel Reiss (University of Idaho)

Fall 2017 (University of Oregon):

• Kirsten Eisenträger (Penn State University) Lecture I, Undecidability in number theory. (video of this lecture)

• Kirsten Eisenträger (Penn State University) Lecture II, Existentially and universally definable subsets of global fields. (video of this lecture)

• Travis Scholl (University of Washington), Isolated elliptic curves in cryptography.

Shabnam Akhtari | University of Oregon |

Derek Garton | Portland State University |

Clay Petsche | Oregon State University |

Holly Swisher | Oregon State University |

Email: oregonnumbertheorydays@gmail.com |