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? How do these vary in families?

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?

Constructing rational curves:
Rational curves are solutions to algebraic equations in rational functions, e.g., the parametrization of the circle above. One of Euclid's postulates is the existence of a line through any pair (p,q) of points; this is the image of the mapping

r(t) = tp + (1-t)q.
Which varieties admit non-constant rational curves? Are there infinitely many and are they dense? Can we find examples passing through prescribed points?