**Cayley-Bacharach theorems and polynomials vanishing on points in projective space**

If Z is a set of points in real n-space, we can ask which polynomials of degree d in n variables vanish at every

point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the

coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A

classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear

conditions are independent? For instance, if we look at the space of lines through three collinear points in the

plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed

by any two of the points imply those of the third. In this talk, I will survey several classical results including the

original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe

some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show

how they connect to several seemingly unrelated questions in modern algebraic geometry.