My research is in the field of algebraic geometry, the geometry of solutions to algebraic equations. A set of such solutions is called an algebraic variety. Here are some specific questions addressed in my work:

Rationality problems:
This is the problem of parametrizing solutions to equations. For instance, the points of the circle satisfy

x²+y²=1 and can be expressed x=2t/(1+t²) y=(1-t²)/(1+t²). Do such parametric solutions exist for more complicated equations, e.g., cubic equations in five variables like v³+w³+x³+y³+z³=1? Are there nice criteria for the existence of parametrizations?

Constructions and birational modifications of moduli spaces:
A moduli space has points corresponding to all the varieties of a given type. For instance, each conic section has an equation of the form

Ax²+Bxy+Cy²+Dx+Ey+F=0, so we can consider (A,B,C,D,E,F) as coordinates on the moduli space of conic sections. Quite generally, moduli spaces satisfy algebraic equations and thus are algebraic varieties. Is there a natural choice for these equations? When there are many possible choices for the equations, how are they related?

Rational points and heights:
Rational points are solutions to algebraic equations in the rational numbers, e.g., (x,y)=(3/5,4/5) is a rational point of the circle. A height is a measure of how large a rational point is. What are the possible heights on a given variety? What natural heights exist on moduli spaces? For which algebraic varieties can we count the points of bounded height?

Geometry and degenerations of plane curves:
Which curves arise as the limits of smooth plane curves? How are these related to singularities of plane curves?