“In particular, the volume functional, which is defining this space, weakly homotopic to r p infinity, should have an infinite number of critical points.”
His hand is parallel to the body, rising and falling in a repetitive movement and moving increasingly away from his chest with each movement before coming to rest at the furthest point. The repetition seems to suggest iterations, an array – his hand cannot do this infinitely, he must reach a point when it seems as though the principle of repetition, of continuation, has been grasped, and then, despite making a specific point about the infinite number of critical points, his demonstrating hand must stop.
Brian Rotman, in his brilliant essay Towards a Semiotics of Mathematics, defines the difference between the mathematician and the fictional entity supposed to execute certain actions within a piece of mathematics, such as taking the sum of an infinite series, which he refers to as the Agent.
Unlike the Mathematician, the Agent is not reflective and has no intentions: he is never called upon to ‘consider’, ‘define’, or ‘prove’ anything, or indeed to attribute any significance or meaning to what he does; he is simply required to behave according to a prior pattern – do this then this then… -imagined for him by the Mathematician. The Agent, then, is a skeleton diagram of the Mathematician in two senses: he lacks the Mathematician’s subjectivity in the face of signs; and he is free of the constraints of finitude and logical feasibility – he can perform infinite additions, make infinitely many choices, search through an infinite array, operate within nonexistent worlds -that accompany this subjectivity.
“So they count as critical points but of course they are not geometrically distinct. So this is a thing we need to rule out.”
Whilst speaking this sentence, Neves casts his eyes quizzically skyward and to the right. Long held to be a sign that someone is lying, this eye movement is more reliably associated with constructed reality or visualisation. Neves here seems to construct a possibility but dismisses it as not as important, or interesting, as he wanted it to be – it’s constructed as a hypothetical, created to be dimissed.
“A priori this just seems some minor or technical issue but understanding why this not gonna happen when we try to find (hand circular motion) this minimal hypersurface, that turns out to be tied down to lots of subtle facts about the topology of these guys. So that’s what I’m gonna try to explain now.”
There’s a phenomenon, and then there’s the significance thereof. A thing can appear to be happening for a simple reason, but reveal some greater structure below. The small circular hand motion occurs when he refers back to the aforementioned minimal hypersurface, and perhaps the quick, circular motion suggests a recycling, a fast-forwarded repitition of the previous discussion.
“So this is the first ingredient of the approach. So let me do now the second ingredient.”
Neves refers explicitly to the ingredients of his recipe, a recipe that brings together elements that act upon one another, puts them through some processes, and creates something distinct at the end. Let’s cook!
“for every k there’s a projective plane that’s homotopic and non-trivial, so we want to make sure that we capture these k-projective planes on this other space. So we’re gonna have a definition to capture that, that’s the notion of k-sweepout.”
At the word “homotopic”, the hand is held parallel to his body, a reference perhaps to the parallel properties of homotopic surfaces. Speaking “k-projective planes” he points to a piece of writing, the chalk scratchings where those planes are held, and then talks about “capture”, including or fixing those things in another conceptual entity, this other space. So the planes are captured in a space using a definition; a set of symbols acts in a conceptual space. The anthropomorphised computer programs of Tron come to mind.
“So the thind ingredient is to define the k with omega k m as we look at all possible k-sweepouts, and then we maximise the volume…”
We have another ingredient here for our recipe but this ingredient is explicitly an action, a decision. It’s simultaneously a substance to be acted upon AND a process, an action.
“So this definition might look a bit e- exotic but there’s um there’s a context from which this was motivated in which we can see this guy uh appear in in in a natural way.”
Exotic is bad. Mathematics is meaningless without the support of the culture around it; Ramanujan’s genius was almost missed because of the strange, unrecognisable notations he used. Mochizuki’s proof of the ABC Conjecture is “just so far removed from what is known and understood by the experts that they have no way of evaluating whether or not he has a new idea that solves the abc problem,” (see article). But, Neves reassures us, this can be framed in such a way that “this guy” appears “in a natural way” – it pops up in a way that seems to us organic rather than artificial, closer to a plant than a factory.
“instead of minimising on the k+1 plane we minimise among k-sweepouts. And then instead of maximising on the k plane we maximise on the particular k-sweepout.”
There’s a strange sort of balancing hand motion with this part – he begins by pointing to one, then another part of the blackboard, and at the moment of talking about maximising, he seems to balance one hand with another in a sort of tipping moment; perhaps this evokes the tipping point of a maximum point, the top of a rollercoaster’s ascent where the balance shifts and the car begins to plunge down again.