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The biggest role is in the effect of the pressure of the skates on the ice melting it to provide a film of water which lubricates the joint between the ice and the skates and lets them glide over the ice.
Then there is the maths regarding the cooling system to ensure the ice stays as ice and doesn't become a big pool of water.
Looking at the actual skating, there is the maths behind the movements. How a skater forces themselves forward; how the skater turns corners, how the skater jumps and does spins.
For example, in doing a spin, there is conservation of momentum: when the skater spins with their arms outstretched they turn slowly, but when they pull their arms in they spin faster: with their arms outstretched, their moment of inertia is larger than when their arms are pulled in. As [rotational] momentum is speed of rotation times moment of inertia, if the moment of inertia reduces, the speed of rotation must increase to keep the same value (as the momentum is conserved) and hence they spin faster.
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