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Differential Geometry

Differential geometry is a broad field of mathematics related and with applications to several areas of mathematics (algebra, analysis, mathematical physics, partial differential equations, topology) and science (biology, chemistry, data analysis, engineering, physics). While topologists have famously been said to be unable to tell the difference between a donut and a coffee cup (since one can be continuously deformed into the other), geometers definitely care about shape. Differential geometry explores geometric quantities such as curvature and volume, including how such quantities evolve or “flow” when one continuously deforms a space using specific geometric constraints. Differential geometry has played an essential role in some of the most difficult mathematical problems in history that, at first glance, seem not to even be problems about geometry. The two most well-known examples are the Poincare Conjecture in topology and Fermat’s Last Theorem in number theory.