Abstract:

The Grothendieck ring of varieties is, depending on your point of view, either a very natural and obvious object to study, or an incredibly strange one with quirks that make it hard to use. I'll give you a definition of the ring and try to show you that it is somehow both very nice and very horrible. This ring has many uses, but one use is in curve counting. I'll end by showing that you can use this ring to come up with a(nother) proof that there are 27 lines on a smooth cubic in P^3, and talk about how you can even refine this to get something better than a number.