Abstract: 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.

This is the second part of this talk.

Abstract: Let $M^n$ be a real $n$-dimensional manifold embedded in $\mathbb{C}^n$. The tangent space of $T_pM$ is totally real at most points $p \in M$. Hence, $M$ is locally polynomially convex at $p$. We may have obstruction to local polynomial convexity at a CR-singularity of $M$. A CR-singularity of order one can be broadly classified as an elliptic or a hyperbolic point. Bishop has shown that $M$ is not locally polynomially convex at an elliptic point $p\in M$. Forstneri\v c and Stout have shown local polynomial convexity of $M$ at $p$ at a hyperbolic point $p\in M^2 \subset \mathbb{C}^2$. We will look at a hyperbolic point $p \in M^n \subset \mathbb{C}^n$ and show local polynomial convexity of $M$ at $p$ under certain condition on the defining functions of $M$.