André Neves – Min-max theory and applications II
Instituto Nacional de Matemática Pura e Aplicada
“I will explain how to find an infinite number of minimal hypersurface in a manifold, OK, or that’s the idea”
His introduction follows the form of an instructional video. How to? The math is already proved, if it’s in a paper in the conference. So the showing of how to is for the audience to create the mathematical meaning in their minds. “That’s the idea” – this addendum suggests the possibility of partial or complete failure. Since the math is already proved, this is perhaps in the sense of understanding by the audience.
He begins with a statement:
“So the theorem is the following right so. Every, uhm [pause, writes (m^(n+2),y)]
…compact Riemannian manifold with so we need two conditions. One is that the dimension is neither too small nor too large, even if this one (gestures to right) is less serious than this one (gestures to left) I’ll explain that shortly
and the second one is that the Ricci, the metric, the metric has positive Ricci curvature
so if I have a manifold that satisfies these conditions then there is an infinite number of minimal embedded hypersurfaces.”
Speech and writing work in parallel in this section. The writing is highly notational, and the speech translates its meaning into more colloquial but much less precise language. The notation is not always read out, for example the symbolic definition that comes before “compact Riemannian manifold”. This may be because it is already known to the audience and is a given, or that it is a specification of which particular instances this applies to which is easier to write than to say. In the former case, this implies that some of the writing done in mathematics is a matter of rigour, included to ensure precision but extraneous to the immediate task of aiding understanding by specific people in a specific context; this piece of writing is for posterity. In the latter case, the fact that some technical definitions are more easily written than spoken attests to the importance given to writing over speaking, also showing an emphasis on recording for posterity. This statement makes considerably more precise the aims spoken at the beginning of the talk.
When speaking about the size of the dimension, Neves gestures to figures that give the upper and lower bounds that seem in this case to define ‘too large’ and ‘too small’, laid out from left to right using arrow notation to explain the relationships between them.
3≤ n+1≤ 7
These symbols, ≤ , are given the meaning ‘less than or equal to’ in mathematics. It means that what’s on the left is smaller than or equal to what’s on the right. As a composite of an arrow, <, and an equals sign, =, it has noticeable iconic as well as symbolic features, defined according to Peirce’s theory of signs. The arrow is composed of two lines that are closer together on the side next to the value indicated to be smaller, and move farther apart toward the side next to the greater value. This widening tallies with a physical experience of larger and smaller objects. The equals sign is written as two lines drawn parallel to one another, indicating equivalence, and could be said to work in a similar way. Simply combining the two takes the reader’s experience of the working of both signs and puts them together to create the new sign, whose meaning can be constructed by this kind of combination. The bottom line is also often drawn as parallel to the bottom of the arrow which brings out this property.
Describing one side of this expression as less “serious” than the other is an interesting move. Of course the scrawled numbers themselves are neither serious, nor frivolous. He seems to be referring to the upper and lower boundaries, suggesting perhaps that one is stricter than the other, but as he says, more on that later.