The deep connection between geometry and physics dates back to ancient times. Applications of calculus, differential equations, and other ideas from analysis to geometry and physics began at the birth of modern science with the foundational works of Newton 350 years ago, and has since continued unabated.

 

The specific subfield of Differential Geometry that is commonly called Geometric Analysis (previously also referred to as global analysis) exploded onto the scene in the 1970s, due in large part to the solution by Shing-Tung Yau of the celebrated Calabi conjecture using methods of elliptic partial differential equations, demonstrating the existence of a particular class of geometric structures (now called Calabi-Yau structures) on compact manifolds satisfying certain conditions. Completely unexpectedly, these Calabi-Yau structures turned out to be exactly the objects required by theoretical physicists in the 1980s to establish the link between abstract string theory and 4-dimensional particle physics.

 

It was also the 1970s that witnessed the explicit reconciliation between geometry and physics, two subjects which had been inextricably intertwined since antiquity but had undergone an estrangement in the 20th century due to the increasing abstraction in pure mathematics. This remarriage between the disciplines was due to interactions between mathematicians such as Atiyah, Bott, Hitchin, Simons, and Singer with physicists such as Yang and Mills, which resulted in the discovery that the mathematical ideas of connections and curvature on vector bundles was essentially identical to the physical ideas of gauge potentials and gauge fields. This realization ignited the area of mathematical gauge theory, leading to spectacular work of Donaldson, Taubes, Uhlenbeck, Yau, and others in the 1980s and beyond.