## Research Interests

I have done work that can be considered complex analysis, number theory, and algebraic geometry. I am especially interested in connections between rational solutions and functional solutions to systems of algebraic equations. For instance, consider the equation of the unit circle,*x*^{2}+*y*^{2}=1. This equation has many rational solutions, such as (3/5)^{2}+(4/5)^{2}=1, coming from Pythagorean triples. The unit circle equation also has the "functional solution" (sin *t*)^{2}+(cos *t*)^{2}=1. On the other hand, if* n*>3, then* x*^{n}+*y*^{n}=1 has only a few rational solutions (this is Fermat's Last Theorem/Wiles's Theorem or the Mordell Conjecture/Faltings's Theorem, depending on what one means by "few" and "rational"). Similarly,*x*^{n}+*y*^{n}=1 has no non-constant "entire function" solutions -- this follows easily from, for instance, the Uniformization theorem. One area I often work in is a field called "p-adic" analysis. Working with functions of p-adic numbers is sort of halfway in between algebra and analysis, so the idea is it might help us understand how functional solutions are related to rational solutions. Another area I work in is Nevanlinna theory, which extends the Fundamental Theorem of Algebra to meromorphic functions. Finally, I have research interests in classical complex analysis, particularly using geometric methods to better understand various inequalities.