My area of interest is Arithmetic Geometry, the use of Algebraic Geometry to solve problems in number theory. Given that number theory itself is maybe best described as a collection of hard problems, it's a pretty wide open area so I will simply describe some things that I've worked with.

Thesis work

My current priority is work towards my thesis, which I will describe here in very down-to-earth terms. Given a polynomial p(x) defined over a field, there is a measure of how symmetric that polynomial is called the automorphism group. If the polynomial is nice enough, the elements of the automorphism group will exactly correspond to the roots of p(x) and in that case we call the automorphism group the Galois group, named after tragic French mathematician Evariste Galois.

Now given a field k, the question can be asked: What possible Galois groups are there over k? That is, if you give me an abstract group, can I give you a polynomial with coefficients in k whose Galois group is the one you gave to me? Certainly that depends on the field! For an algebraically closed field like the complex numbers, the only possible Galois group is {id}, the identity, because the only irreducible polynomials have degree 1. Likewise over the real numbers there are only two possible Galois groups, {id} and {id, complex conjugation} because an irreducible polynomial over the reals has degree only one or two. On the other hand, over the rational numbers the answer is much more complicated and in fact not known in general. There is not a single group which is known to NOT occur over the rational numbers. Meanwhile tons of exotic groups are known to occur. Though data entry methods prohibit us from writing down the actual polynomial(it would have degree roughly 8*10^{53}) there is known to be a polynomial which has the Monster Simple Group as its Galois Group. For a fantastic exposition for why this is true, see the following link.

The Inverse Galois problem is the conjectural statement that in fact every possible finite group appears as a Galois group of some polynomial over the rationals. The Regular Inverse Galois problem is the same, but for the field of rational functions (that is, ratios of polynomials) over the rational field, with some extra structure. It is deceptively hard to coerce groups to appear, and one fruitful way to tackle both problems at once has been the method of Shih to induce certain geometric objects called modular curves to give projective matrix groups over finite fields as Galois groups. For an introduction to this material, please see the slides I used for my oral exam: http://www.math.uga.edu/~jstankew/ShihTalk.pdf. Please excuse the density of the slides, as there was a lot of material to get through and I wanted to take the pauses out for the web.

The Frobenius Problem

In Fall 2008 I assisted Prof. Dino Lorenzini in his IVRG(Introductory Vigre Research Group, the participants were undergraduate students at UGA) on the Frobenius Problem, aka the postage stamp problem. The classical question is: if we have postage stamps of n different, relatively prime integral values (e.g. 2,7 and 25 cent stamps) what is the greatest number which cannot be represented? To push it further, what is the greatest number represented exactly once? twice? thrice? The results of our exploration are found in this preprint. In Spring 2010 I continued this work, finding (in joint work with Jeffrey Shallit at the University of Waterloo) that g₀ - g_k can be arbitrarily large.

Torsion on Elliptic Curves

My first research experience was in the Number Theory VIGRE Research Group. Our object of study has been the study of Torsion Subgroups of CM Elliptic Curves. That is, what can we say about the groups which appear as torsion subgroups of elliptic curves?

Moduli of Points on Curves

A common theme to most of the topics above is the use of moduli spaces. Recently I've had occasion to work on a project dealing with divisors on M_{0,n}, the space of curves of genus 0 with n marked points which is very closely tied to the moduli of hyperelliptic curves. More to come on this.