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

x^{2}+y^{2} = 1
and can be expressed
x = 2t/(1+t^{2})
y = (1-t^{2})/(1+t^{2}).
Do such parametric solutions exist for more complicated
equations, e.g., cubic equations in five variables like
v^{3}+w^{3}+x^{3}+y^{3}+z^{3} = 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

Ax^{2}+Bxy+Cy^{2}+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?