“*…and we put there two coefficients because we don’t care about the orientation, a surface – a hypersurface or the same hypersurface with different orientation, they are the same object.”*

This sentence is accompanied by a flipping of the hand from right to left and back, palm up to palm down and back. This suggests flipping around a particular axis, the hand becoming the hypersurface, orientation exemplified by the idea of upsidedownness or rightwayupness. This representation of the idea he is trying to communicate draws out the interesting assertion that “they are the same object”, rather than two objects with the same form.

“*So what the theorem is saying is that in some cases we can make sure that this function will have an infinite number of distinct critical points, OK. So, so let me explain that part, because the whole issue hinges on a fact which is <seems subtle?> a priori but actually is tied down to very big facts about the structure of that space. So let me explain that now.”*

Having written down a lot of expressions Neves stops and steps away to offer an interpretation, to draw out the important implications of these equalities. The symbols are an inert set of objects, like a pile of stones that lie in relationship to one another, oblivious to the aims and desires of the humans using them as steps to ascend a steep face.

“*S**o from the basic principle of Morse Theory or Min-Max Theory as Fernando explained if I wanna have an infinite number of, uh, critical points then I must have a lot of topology there, right.”*

The idea of an area having a lot of topology is surprising to me. I can imagine a textbook having a lot of topology, or a person learning a lot of topology, but I would expect only to be able to talk about a quantity of study of a thing in a meta-mathematical context.

**1. ****1**.

Mathematics

the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.

- a family of open subsets of an abstract space such that the union and the intersection of any two of them are members of the family, and that includes the space itself and the empty set.
plural noun:

**topologies** **2**.the way in which constituent parts are interrelated or arranged.

“the topology of a computer network”

The phrase uses ‘topology’, not ‘topologies’, so it doesn’t refer to a number of families.

Perhaps instead this phrasing suggests a quantity of study in a different sense, a quantity of material for study, an area with complex topology.