Are lines even real?

Fix four pieces of string, tied to opposite ends of the room. How many other pieces of string can you stretch across the room which touch the initial four you started out with? The 19th century enumerative geometer will tell you there are exactly two pieces of string satisfying this property, i.e. there are two lines meeting four lines in three-space. This, unfortunately, is a lie, because by a “line” this geometer means a copy of the complex projective line, rather than a literal string. We will discuss conditions where this lie is no longer a lie – that is, when solutions to problems of this flavor are actually defined over the reals. In low dimensions we need no tools more complicated than linear algebra, although in higher dimensions, methods derive from all corners of mathematics: from topology to algebra to analysis. Time pending we will discuss recent explosive progress in this area in the past few decades, and a suite of conjectures that guide current thinking about “reality” in enumerative geometry.