By Luke Keating Hughes
The idea of de Rham’s Theorem is that with two very different approaches to making measurements on a space we can find striking similarities. The spaces we are talking about locally look like flat space but can look very different when we zoom out. For example, the world looks pretty flat as we walk around, but if we keep walking in one direction for long enough we will end up arriving in the same place we started. This tells us that although locally the surface of the earth is flat, globally it is not as similar to flat space as we might originally have thought.
Imagine we have three French ants: Élie, Henri and Georges. Élie, Henri and Georges have found some Parisians sitting down to a picnic and decided to wander onto an item of food our picnic-goers have left lying on the ground. It’s getting close to lunch time and our three French ants are starting to get hungry! Unfortunately, Georges is a celiac and would like Élie and Henri to check whether or not they are on a donut. Élie thought it looked like a donut from afar, which would be unfortunate for Georges, but Henri could have sworn it was an apple! Up close it is impossible to tell because to an ant it all just looks like flat space, so Élie, who just happens to have brought a windsock with him, exclaims “Oh, I will take measurements about the speed and direction of wind passing over the food and deduce its shape based upon that!” Now Henri, who has brought a bag of little flags with him, pipes in, “Élie, that will take too long! Instead, I will place these flags all over the food and deduce the shape by where the flags lie.”. Our ant Georges is a smart guy and realises that in fact at a very deep level these ants are actually calculating the same thing, just in very different ways.
At the heart of this, Georges is actually talking about de Rham’s Theorem. By calculating the speed and direction of wind Élie is forming what a mathematician would refer to as a vector field on the surface of our food, belonging to the field of differential geometry. On the other hand, Henri is taking note of the shape of the surface of the food, how the different parts of the shape interact with each other, belonging to the field of topology. These two very distinct approaches come to a profound cross roads when we talk about the shape of a space and Georges can be safe in knowing that he will not have bowel distress.
Luke Keating Hughes was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.