Syzygies interpolate between torsion-freeness and freeness. In this talk, we will introduce the concept of syzygies and review criteria for a module to attain a certain syzygy order. Then we will discuss a result of Franz which relates the syzygy order of equivariant cohomology of a toric variety to the combinatorics of the underlying fan. Finally, by restricting the torus action on toric varieties to subtori, we will investigate the resulting changes in the syzygy order of their equivariant cohomology.

A compact subset K of C^n is said to be polynomially convex if every point in the complement of K can be separated from K by a polynomial. Oka-Weil theorem tells us that holomorphic functions can be approximated by polynomials on polynomially convex sets. This is a generalization of Runge approximation theorem to higher dimension. We will go through some approximation results in complex analysis, look at examples of polynomially convex sets, and analyze the local polynomial convexity of real n-dimensional manifolds embedded in C^n.