Purpose: The goal of this seminar is to understand Galois-theoretic aspects of maps between curves, starting with Belyi's theorem that on any curve defined over a number field there is a nonconstant rational function which has at most three branch points.
Meetings:
| 1/26/11 | Mike Zieve | Belyi's theorem |
| 2/2/11 | Mike Zieve | Refinements of Belyi's theorem |
| 2/9/11 | Alex Mueller | Descent, I |
| 2/16/11 | Sijun Liu | abc implies effective Mordell |
| 2/23/11 | Mike Zieve | Descent, II |
| 3/30/11 | Mike Zieve | The converse of Belyi's theorem |
| 4/6/11 | Zach Scherr | Preimages of branch loci of Belyi maps |
| 4/13/11 | Alex Mueller | Dessins d'enfants |
| 4/20/11 | Julian Rosen | Geometric Galois actions |
| 5/12/11 | Zach Scherr | Belyi's theorem in positive characteristic |
Participants: Hunter Brooks, Sijun Liu, Alex Mueller, Julian Rosen, Zach Scherr, Ben Weiss, Jeremy West, Mike Zieve
Summer 2010
Purpose: The goal of this seminar is to explain background material used in the research projects of Alex Carney, Ruthi Hortsch, and Ben Weiss. This includes topics from algebraic topology, Riemann surfaces, group theory, Galois theory, algebraic geometry, elliptic curves, height functions, etc.
Lecture notes: Here are some unpolished lecture notes. I (M.Z.) would welcome any feedback!
Meetings:
| 6/24/10 | Mike Zieve | Riemann's existence theorem |
| 6/28/10 | Alex Carney and Ruthi Hortsch | Polynomials and group theory |
| 7/15/10 | Mike Zieve | Primitive and multiply transitive permutation groups |
| 7/21/10 | Mike Zieve | Generic maps between curves |
| 8/10/10 | Mike Zieve | Hurwitz spaces |
| 8/18/10 | Mike Zieve | Ramification and Puiseux series |
| 8/26/10 | Mike Zieve | Fibered products |
Participants: Hunter Brooks, Alex Carney, Nic Ford, Ruthi Hortsch, Jeff Lagarias,
Dylan Moreland, Alex Mueller, Julian Rosen, Zach Scherr, Ari Shnidman, Ben Weiss, Mike Zieve
Winter 2010 (b)
Purpose: The goal of this seminar is to introduce the various concepts and results which will be used in Matt Baker's Michigan Lectures in Number Theory.
Meetings:
| 4/2/10 | Alex Mueller and Julian Rosen | The Berkovich projective line |
| 4/9/10 | Sijun Liu and Zach Scherr | Canonical heights and Bilu's equidistribution theorem |
| 4/16/10 | Hunter Brooks and Ari Shnidman | Green's functions and the Mandelbrot set |
Participants: Hunter Brooks, Sijun Liu, Alex Mueller, Kartik Prasanna, Julian Rosen,
Zach Scherr, Ari Shnidman, Ben Weiss, Mike Zieve
Winter 2010 (a)
Purpose: The first goal of this seminar is to understand the proof of the Ax-Katz theorem, which gives a lower bound on the power of p dividing the number of points on certain affine varieties over Fpn. The second goal is to understand Alex Mueller's improvement of the Ax-Katz bound, and to examine whether further improvements are possible.
Meetings: Alex Mueller led meetings on 3/22/10 and 3/29/10.
Participants: Sijun Liu, Alex Mueller, Julian Rosen, Zach Scherr, Mike Zieve
Fall 2009
Purpose: The goal of this seminar is to help students gain working knowledge of class field theory by using it to solve some concrete problems. As a bonus, it will likely lead to one or more publishable papers.
Meetings: Meetings involve a combination of student lectures on background material and group discussion of aspects of the problem. The topics and discussion leaders are:
| 11/12/09 | Mike Zieve | Motivation and introduction to the problem |
| 11/19/09 | Alex Mueller | Decomposition of places in function field extensions |
| 11/24/09 | Julian Rosen | The Carlitz module |
| 12/4/09 |
All | Examples |
Topic: Find abelian extensions of Fq(x)
having more degree-one places than any other
known function fields of the same genus over the same constant field. More
generally, analyze which combinations of conductor and splitting places
yield the best function fields of this type. If there is time and interest,
consider the same problems for towers of abelian extensions of function fields.
Participants: Alex Mueller, Hieu Ngo, Julian Rosen,
Zach Scherr, Ari Shnidman, Sun Xinyun, Ben Weiss, Mike Zieve