Gromov-Witten Theory of P^1xP^1xP^1

We prove equivalences between the Gromov-Witten theories of toric blowups of P^1xP^1xP^1 and P^3. In particular, we prove that the all genus, virtual dimension zero Gromov-Witten theory of the blowup of P^3 at points precisely coincides with that of the blowup at points of P^1xP^1xP^1, for non-exceptional classes. We also prove there exists a symmetry of the Gromov-Witten theory of P^1xP^1xP^1 analogous to and intimately related to Cremona symmetry of P^3. Finally, we discuss enumerative applications of these results. This is joint work with Dagan Karp.