**Title**: One Good Reason a Topologist Cares about Algebraic Geometry

**Abstract**: Why would a topologist be interested in the zero set of a complex polynomial like *f*(*x*, *z*) = *x*^{2}(2 – *z*) + (*z*^{2} – *z* – 1) or in the points at infinity of its compactification? Polynomials like these arise as canonical components of character varieties of 1-cusped hyperbolic 3-manifolds and the points at infinity detect essential surfaces living inside the manifold. Such surfaces lend insight into the topology of the manifold and understanding their structure is interesting. The detection of these essential surfaces via character varieties is attributed to work of Marc Culler and Peter Shalen in the 1980s. In my talk I will describe this machinery, present concrete examples, and discuss some current research.