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Series: Algebra Seminar

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 $\phi$.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.)

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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 $p > 0$.

Series: Algebra Seminar

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.

Series: Algebra Seminar

Polytropes are both ordinary and tropical polytopes. Tropical types of
polytropes in \R^n are in bijection with certain cones of a specific
Gr\"obner fan in \R^{n^2-n}. Unfortunately, even for n = 5 the entire fan
is too large to be computed by existing software. We show that the
polytrope cones can be decomposed as the cones from the refinement of two
fans, intersecting with a specific cone.
This allows us to enumerate types of full-dimensional polytropes for $n =
4$, and maximal polytropes for $n = 5$ and $n = 6$.
In this talk, I will prove the above result and describe the key
difficulty in higher dimensions.

Series: Algebra Seminar

Similar to the glamour of Las Vegas, the excitement and drama of winning in casinos and falling under the spell of such legends as Frank Sinatra and Dean Martin; is the search for revealing the mystery of absolute Galois groups and their special properties among other profinite groups. The recent, spectacular proof of the Bloch-Kato conjecture by Rost and Voevodsky, with Weibel's patch, and some current and interesting developments involving Massey products, hold great promise and new challenges on the road to understanding the structure of absolute Galois groups. This talk will provide an overview of the subject, and then explain some recent results obtained with Nguyen Duy Tan.