Scholarship

My research sits squarely in between algebra, the study of polynomial equations and their solutions, and geometry, the study of shapes and their structure. Polynomials are formed using only the operations of addition and multiplication; they are sums and products of powers of variables, e.g. 3x3 + 4y3 + 5z3. In our calculus courses, we learn that to approximate values of complicated functions that might arise, we can often use polynomials whose values are much easier to compute. For example, one can show that sin(x) ≈ x when x is close to 0, so we may approximate sin(0.1) by 0.1 (for comparison, the true value is 0.09983...). 

Many familiar objects are described by polynomial equations, formed by taking a polynomial and setting it equal to 0. For example, a circle of radius 1 can be defined by taking the collection of all solutions to the equation x2 + y2 − 1 = 0. This equation describes the circle algebraically, but alternatively we may describe it geometrically by graphing.

Both the algebraic and geometric description are valid ways of describing the circle. The geometric picture often allows us to “see” structure such as symmetry that an equation alone cannot capture as well. So it can be useful to have a dictionary that allows us to go back and forth between algebra (the equation) and geometry (the shape and its properties). We need not restrict ourselves to graphs of curves in two dimensions, however. Here the “two” comes from the two directions: up/down and left/right. A sphere in our usual 3-dimensional world can be described either by (solutions to) the equation x2 + y2 + z2 − 1 = 0 or by its graph. In my own work, I consider equations defining surfaces called K3 surfaces, such as xyz(x + y + z) − 1 = 0, which is modeled graphically by the colorful image above.

This picture hints at the kinds of complexities found in this sort of surface, especially in comparison to a circle or a sphere. 

Sometimes only certain kinds of solutions are relevant or even make sense for a particular equation. For example, when methane gas burns, methane (CH4) combines with oxygen (O2) to form carbon dioxide (CO2) and water (H2O). The number of oxygen atoms involved in the reaction can be described by the equation 2y = 2w + z. But, y, w, and z must be positive integers since they represent the number of O2, CO2, and H2O molecules, respectively. Polynomial equations where only integer solutions are considered are called Diophantine equations. 

My research often focuses on either determining when various Diophantine equations have (integer) solutions, or providing a definitive explanation for why no solutions can exist. In the equation for a sphere, it is not too hard to spot the solution x=1, y=0, z=0. To prove an equation has no solutions, however, we often draw insight both from the algebraic and geometric structure. For example, it turns out that the equation xyz(x + y + z) − 1 = 0 has no integer solutions, which is not as easy to see. If we try to make the problem simpler by only looking for solutions where x, y, z can be 0, 1, or 2, ("modulo 3" wherein we only keep track of remainders after division by 3 ) through trial and error, we quickly find that none exist. In this case (with a bit of abstract algebra) this is enough to explain the lack of solutions. Most of the time we are not so lucky, and then we turn back to the geometry to guide us, e.g. by trying to use properties of symmetry to help narrow down where to search for solutions. Much of my work is based on this interplay between algebra and geometry.