This is not meant to be a fully convincing answer, but to argue for one possible way that pure mathematics research is important. I believe that mathematics has some special features that make it good for promoting international cooperation and friendship, building bonds between diverse peoples.

First, as is clear to us who do it, pure mathematics is interesting all on its own. One manifestation of this is that it doesn't seem to depend on cultural background or political ideology whether the ideas in mathematics are found to be interesting by a sufficiently curious mind. The same ideas can be appreciated regardless of one's overall perspective.

Second, mathematics is (pretty much) objective. One can recognize the value of a mathematician's contribution, and if one has the bare minimum of honesty, it will trump all other prejudgements about the other party. In this way, it has the ability to break through prejudice or adversarial political commitments, without first requiring subjective emotional change to get started.

Third, mathematics only requires time and thought. In modern times, once mathematicians with close interests are introduced, collaboration can begin easily and will not necessarily require costly investments by governments or others to get going, in contrast to experimental science.

Of course, all the pure intellectual pursuits of mankind are valuable in themselves. But mathematics, though practiced by a small number of us, seems to have a special ability to transcend cultural barriers and perhaps thus contribute in some small way to peace.