1. André Neves – Min-max theory and applications II PART TWO

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Remarks. One is that if n plus two equals one, the theorem is false. And the reason is because of uh one word I forgot to write which is. Then there’s an infinite number of minim- oh yeah I wrote it. OK. Then the reason is because of this word. [draws a square around ’embedded’]”

Another metonym. Those scrawled letters have no direct effect on the mathematics, but they are what he indicates, with an accusatory box, as the reason.

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Franks and Bangert in 92 they showed that every two-sphere admits an infinite number of closed geodesics. Of course, they’re gonna be badly immersed and they’re gonna have lots of self-intersection.”

This math is tied to 1992.


Every two-sphere admits closed geodesics, allowing them in. A two-sphere is basically a sphere as we know it, a two-dimensional surface that wraps around to create the form of a ball; a geodesic is essentially a straight line in curved space.

“Spherical triangle” by derivative work: Pbroks13 (talk)RechtwKugeldreieck.svg: Traced by User:Stannered from a PNG by en:User:Rt66lt – RechtwKugeldreieck.svg. Licensed under Public Domain via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Spherical_triangle.svg#mediaviewer/File:Spherical_triangle.svg

Admit” suggests a confession, a compromise, welcoming somebody into your house. The two-sphere, a definition defining a surface, now allows itself to be defined in a different way, as a thing that contains an infinite number of closed geodesics. The definition gives rise to a conceptual entity that then re-evaluates itself in the phrasing of its description. The word “immersed” suggests a smaller thing engulfed by a larger thing, a person immersing themselves in a swimming pool.

They’re gonna be badly immersed”

Quality in mathematics is a hotly-disputed question. What is good math, and what is bad? For it even to count as mathematics, it must be correct. G. H. Hardy said that a good piece of mathematics needed the qualities of “unexpectedness, combined with inevitability and economy”.

“A mathematical proof should resemble a simple and clear-cut constellation, not a galaxy in the milky way” – Hardy, G. H., A Mathematician’s Apology, Cambridge: Cambridge University Press, 1992 (first published 1940)

Conversely, bad or ugly mathematics might be the less neat, the sprawling. This tallies with AI expert Juergen Schmidhuber’s articulation of aesthetic experiences.

“Among several patterns classified as “comparable” by some subjective observer, the subjectively most beautiful is the one with the simplest (shortest) description, given the observer’s particular method for encoding and memorizing it” – Schmidhuber, Jürgen, “Theory of Beauty and Low-Complexity Art”, Jürgen Schmidhuber’s Home Page, 1997 < http://www.idsia.ch/~juergen/&gt;

The geodesics being “badly” immersed is associated with “a lot of self-intersection”. Perhaps this is described as badly immersed because those self-intersections are somehow uncomfortable for the human mind.


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