Physics and Mathematics have been married, divorced and remarried in what must be one of history’s most remarkable intellectual relationships. We now seem to be in a period of relative bonhomie spurred by a mutually enriching give and take of ideas in the last few decades. A particularly fruitful arena for this exchange has been in quantum field theory, particularly gauge theories, and relatedly string theories.

In fact, a major engine of theoretical developments in physics has been what is called Gauge-String Duality (aka the AdS/CFT correspondence). I will try to convey not only the physical richness of this unexpected duality (a bit like the Langlands Correspondence in scope and ambition) but also its considerable mathematical ramifications. I will illustrate the latter through specific instances of increasing complexity, which have already proved to be mathematically significant. Finally I will outline a broad program which aims to derive this duality from first principles using some striking mathematical results on Riemann surfaces. I will not assume much knowledge of physics and try instead to convey the broad sweep of the ideas.