Research Statement – Rafe Jones

Arithmetic and Diophantine Geometry

Responsible Vouchering in Turtle Research: An Introduction and Recommendations

1Biodiversity Alliance, c/o Cleveland Metroparks Zoo, Cleveland, Ohio 44109 USA [[email protected]]; 2Institute of Biodiversity and Environmental Conservation, Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia [[email protected]]; 3Department of Biology, Texas State University, San Marcos, Texas, 78666 USA [[email protected]]; 4Natural History Museum, Biodiversity Re...

A Postmodern View of Fractions and the Reciprocals of Fermat Primes

In America’s visually-oriented, quantitatively illiterate culture, images have a great deal of power, so if a picture is today worth a thousand words, it must be worth a least a billion numbers. This power of the image is a hallmark of the postmodern era, in which the critical role of the observer has come to be recognized, and an understanding of the viewpoint has become inseparable from that ...

Arboreal Galois Representations

Let GQ be the absolute Galois group of Q, and let T be the complete rooted d-ary tree, where d ≥ 2. In this article we study “arboreal” representations of GQ into the automorphism group of T , particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of GQ. We first give some methods of constructing arboreal rep...

Achievement Sets of Sequences

Given a real sequence (xn), we examine the set of all sums of the form P i∈I xi, as I varies over subsets of the natural numbers. We call this the achievement set of (xn), and write it AS(xn). For instance, AS(1/2) = [0, 1] by the existence of binary expansions, and AS(2/3) is the Cantor middle third set. We explore the properties of these two sequences that account for their very different ach...