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:
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?
This is the problem of parametrizing solutions to equations.
For instance, the points of the circle satisfy
Constructions and birational modifications of moduli spaces:
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
Is there a natural choice for these equations?
When there are many possible choices for the equations,
how are they related?
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
Constructing rational curves:
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?
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