Posted: Mar 21, 2019 9:05 am
Taken from here.
Calculus and then the calculus of variations seems a sensible route into this stuff. I recall studying the building of tunnels (e.g. under the Mersey) - minimum path and dug materials - as a calculus of variations problem in undergraduate maths at Liverpool.
Karen Uhlenbeck, Uniter of Geometry and Analysis, Wins Abel Prize
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To understand what it means for a map to be harmonic, imagine some compact shape made of rubber — a rubber band, say, or a rubbery sphere. Next, choose a particular way to situate this shape inside a given space (such as an infinite three-dimensional space or a three-dimensional doughnut shape). This positioning of the shape is called a harmonic map if, roughly speaking, it puts the shape in equilibrium, meaning that the rubber won’t snap into some different configuration that has lower elastic potential energy (what mathematicians call Dirichlet energy).
When the space you’re mapping the rubbery shape into is a complicated object with holes (such as a doughnut surface or its higher-dimensional counterparts), a variety of harmonic maps may emerge. For instance, if you wrap a rubber band around the central hole of a doughnut surface, the band cannot shrink all the way down to a point without leaving the surface of the doughnut — instead it will contract down to the shortest route around the hole.
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Calculus and then the calculus of variations seems a sensible route into this stuff. I recall studying the building of tunnels (e.g. under the Mersey) - minimum path and dug materials - as a calculus of variations problem in undergraduate maths at Liverpool.