How many solutions does the equation $x^2+y^2 = z^2$ have if $x$, $y$, $z$ are taken from $Z/pZ$? How many square-free polynomials of degree $n$ are there with coefficients in $Z/pZ$? Questions of this kind have deep and seemingly unexpected connections with the arithmetic and geometry of algebraic varieties. Projects will involve an exploration of this connection. They can range from heavily theoretical to mostly  experimental, for example, involving computer programming to do these counts, and using the data to make some conjectures.