Toric varieties are a particularly well-behaved class of algebraic
variety, almost but not quite in 1:1 correspondence with convex polyhedra.
Many of them (the interesting ones?) are also symplectic manifolds.
Any question one asks about a toric variety (e.g., the cohomology ring)
comes down to a combinatorial question; conversely, many of the deeper
results about convex polyhedra have been proven by constructing the
associated toric varieties and turning the combinatorial problems
into better-studied geometric ones. In this talk I'll show how to
construct these spaces using basic differential topology, as a sort of
first interesting example of symplectic and algebraic manifolds.