## Seminars and Colloquia by Series

### Inclusion of Spectrahedra, the Matrix Cube Problem and Beta Distributions.

Series
Algebra Seminar
Time
Monday, March 9, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor KlepUniversity of Auckland
Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based onmatricial’’ spectrahedra and estimate the error inherent in this relaxation. The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.

### Uniform bounds on rational points on curves of low Mordell-Weil rank

Series
Algebra Seminar
Time
Wednesday, February 18, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eric KatzUniversity of Waterloo
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves. I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform. These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.

### An Equidistribution Result in Non-Archimedean Dynamics

Series
Algebra Seminar
Time
Monday, January 26, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kenny JacobsUniversity of Georgia
Let K be a complete, algebraically closed, non-Archimedean field, and let $\phi$ be a rational function defined over K with degree at least 2. Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of $\phi$. The first is a function $\ord\Res_\phi$, which describes the behavior of the resultant of $\phi$ under coordinate changes on the projective line. The second is a discrete probability measure $\nu_\phi$ supported on the Berkovich half space that carries arithmetic information about $\phi$ and its action on the Berkovich line. In this talk, we will show that the functions $\ord\Res_\phi(x)$ converge locally uniformly to the Arakelov-Green's function attached to $\phi$, and that the family of measures $\nu_{\phi^n}$ attached to the iterates of $\phi$ converge to the equilibrium measure of $\phi$.​

### Quadratic points on hyperelliptic curves

Series
Algebra Seminar
Time
Friday, November 21, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jennifer ParkMcGill University
Using the ideas of Poonen and Stoll, we develop a modified version of Chabauty's method, which shows that a positive proportion of hyperelliptic curves have as few quadratic points as possible.

### Effective Chabauty for symmetric powers of curves

Series
Algebra Seminar
Time
Wednesday, November 19, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jennifer ParkMcGill University
Faltings' theorem states that curves of genus g> 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry and tropical geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

### Moduli of Tropical Plane Curves

Series
Algebra Seminar
Time
Friday, November 14, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Ralph MorrisonBerkeley
Smooth curves in the tropical plane correspond to unimodulartriangulations of lattice polygons. The skeleton of such a curve is ametric graph whose genus is the number of lattice points in the interior ofthe polygon. In this talk we report on work concerning the followingrealizability problem: Characterize all metric graphs that admit a planarrepresentation as a smooth tropical curve. For instance, about 29.5 percentof metric graphs of genus 3 have this property. (Joint work with SarahBrodsky, Michael Joswig, and Bernd Sturmfels.)

### Joint Athens-Atlanta Number Theory

Series
Algebra Seminar
Time
Tuesday, November 4, 2014 - 16:00 for 4 hours (half day)
Location
Skiles 005
Speaker
Arul Shankar and Wei ZhangHarvard University and Columbia University
The Joint Athens-Atlanta Number Theory Seminar meets once a semester, usually on a Tuesday, with two talks back to back, at 4:00 and at 5:15. Participants then go to dinner together.

### Exceptional isogenies between elliptic curves

Series
Algebra Seminar
Time
Monday, November 3, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
François CharlesMIT and Paris-Sud
We will discuss a proof of the following result: if E and E' are two elliptic curves over a number field, there exist infinitely many places p of k such that the reduction of E and E' modulo p are isogenous. We will explain the relationship with the dynamics of Hecke correspondences on modular curves and the heuristics behind such results.

### F-singularities and weak ordinarity

Series
Algebra Seminar
Time
Monday, October 20, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Karl SchwedeUniversity of Utah
I will discuss recent work of Bhargav Bhatt, myself and Shunsuke Takagi relating several open problems and generalizing work of Mustata and Srinivas. First: whether a smooth complex variety is ordinary after reduction to characteristic $p > 0$ for infinitely many $p$. Second: that multiplier ideals reduce to test ideals for infinitely many $p$ (regardless of coefficients). Finally, whether complex varieties with Du Bois singularities have $F$-injective singularities after reduction to infinitely many $p > 0$.

### Economics for tropical geometer

Series
Algebra Seminar
Time
Tuesday, October 7, 2014 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ngoc Mai TranUT Austin
This talk surveys the connection between economics and tropical geometry, as developed in the paper of Baldwin and Klemperer (Tropical Geometry to Analyse Demand). I will focus on translating concepts, theorems and questions in economics to tropical geometry terms.