After reviewing some results from last week's talk concerning moment-angle complexes $\mathcal{Z}_K$ and their cohomology rings, I will describe some further structure on $H^*(\mathcal{Z}_K)$ given by cohomology operations induced by the standard torus action. Under the identification of $H^*(\mathcal{Z}_K)$ with the Koszul homology of the Stanley-Reisner ring of $K$, these operations assemble to give an explicit differential on the minimal free resolution of the Stanley-Reisner ring. Using this topological interpretation of the minimal free resolution, we give simple algebraic and combinatorial characterizations of equivariant formality for torus actions on moment-angle complexes. This is joint work with Benjamin Briggs.

A compact set $X\subset\mathbb C^n$ is said to be rationally convex if for every point $z\not\in X$ there is a polynomial $P$, depending on $z$, so that $P(z)=0$ but $P^{-1}(0)\cap X=\varnothing$. In view of the Oka-Weil theorem, any function holomorphic on a rationally convex compact $X$ can be approximated uniformly on $X$ by rational functions with poles off $X$. A totally real manifold $M$ is one whose tangent space has no complex structure, i.e., $J(T_pM)\cap T_pM=\{0\}$ for all $p\in M$. $$ $$ By a classical result of Duval-Sibony, a totally real manifold $M$ in $\mathbb{C}^n$ is rationally convex if and only if there exists a Kähler form $dd^c\varphi$ for which $M$ is isotropic. Under a mild technical assumption, we generalize this necessary and sufficient condition to the setting of totally real sets (zero loci of strictly plurisubharmonic functions).

Numerical algebraic geometry is a computational framework for studying solution sets to polynomial equations (called varieties) using numerical algorithms. In contrast to symbolic methods (e.g. Grobner bases) which manipulate polynomials, numerical algorithms manipulate points on varieties. In this sense, numerical algebraic geometry may be thought of as the geometric side of computational algebraic geometry. In this talk, I will explain the basics ideas underlying the theory of numerical algebraic geometry.