Mathematics of the topological open string

A major focus of modern mathematical research is in giving justification to predictions made by nonrigorous physical reasoning. A typical example: a physicist argues that some quantity must remain constant as one varies from one regime of some physical system to another, but on the other hand can be
computed in very different ways in the different regimes. Frequently in the cases of mathematical interest, such reasoning is made in the context of quantum field or string theory, and therefore cannot be imported into mathematics because these theories are not mathematically rigorous, in particular because the ‘path
integral’ remains undefined almost a century after its appearance. Nevertheless the calculations in the limiting regimes often can be put on sound mathematical footing, and in these cases, mathematicians have expended great effort to prove that, in fact, the physicist’s predictions were correct.