Having referred to “Yau, 82” as the reason they were pursuing this work, Neves says:
So we have I think yes two or three versions of this conjecture. He restate this conjecture in two or three surveys. In the first one I think it’s the first problem in the minimal surface section in the first problem he gave he states immerse and then later on he asks for just a number of minimal embedded surfaces. I think the reason he puts immersed here is because he knows that just on s2 with any metric the result is false, OK”
This passage sounds more like archaeology than mathematics; he is unpicking the intentions and thoughts of a predecessor using the incomplete information provided by their writing. The difference between “embedded” and “immersed” also comes to the fore. The presentation and a little (uninformed) reading suggests that “embedded” is a more demanding term than “immersed” – an immersed object is allowed to self-intersect, whereas for something to embed into something else, it would have to fit inside with that number of dimensions without self-intersecting. The distinction between these two words is interesting; dealing as we are with conceptual space of varying numbers of dimensions, the terms do not mean the same thing as in the physical, 3-dimensional world. The word “embedded” suggests a tight, solid fit, a piece of shrapnel embedded in an oak door. “Immersed” remains fluid, a casual, easy dip in.
“If m3 is hyperbolic then this is true by Khan-Markovic this was in 2012 or Rubinstein in 2005”
Brian Rotman describes the narratives implied by mathematical writing as combining the actions of the implied Subject, to whom the directions to consider, add etc. are directed, the Agent, imagined to do impossible work such as calculating an infinite sum, and the Person, the reader of the text who constructs mathematical meaning in their mind. He identifies assertions of truth and existence as using a language of prediction: “In making an assertion the Subject is claiming to know what would happen if the sign activities detailed in the assertion were to be carried out.” – Brian Rotman, Mathematics as Sign: Writing, Imagining, Counting, Stanford University Press, 2000
This phrasing in particular is interesting: this is true by Khan-Markovic”
See my blog entry about people’s names and terminology: https://infiltratemathematics.wordpress.com/2014/11/04/names-naming-and-terminology/
To say that it is trueby Khan-Markovic” gives those names the air of a tool, or a route, their work becoming one of those two things.
Going on to describe Khan-Markovic and Rubinstein’s different proofs, he explains the different routes taken than show the same thing; there must be some advantage gained to outlining both. Perhaps each suits different minds, or specialisms. He mentions that Rubinstein he has a different method, he looks at Hagar splittings of many covers of m3. But Rubinstein, he outlines this very carefully but he does need to use the index estimate, which as Fernando said has not been proven yet.” Though mathematics is seen as such a precise art, governed by fact above human desires, it is sometime very utilitarian; if a thing is believed to work, but for some reason has not been concretely proven, people will continue to use it, albeit highlighting this fact.
Having filled the three boards with the theorem and remarks, Neves moves away from it, beginning to discuss “What I will explain now is how to find these um minimal um hyp hypersurface so in in our theorem we are as far from hyperbolic as possible and usually in these type of problems there’s a big difference between this hyperbolic world in which it’s very rich in topology and so you find the geodesics are the minimal surface by finding lots of topology in (??) manifold or the case where you have positive Ricci or your manifold is a sphere in which case we have no topology at all and so if you wanna find minimal hypersurface you have to look at the space of all hypersurface in a given manifold”
He begins with “I”, then uses “we”, and finishes with “you”. The “I” refers to his activity as an explainer, obviously more clearly identified with himself as an individual. The “we” may be referring to his work in collaboration on the paper, in which case the speech starts to focus more on the written/proven mathematics, or perhaps it may be moving to include the audience (in the room or public). There is a later instance, “in which case we have no topology at all”, which seems like more of a public sort of “we”, referring perhaps to the mathematics community. The properties of this object in a sense belong to the mathematics community. When he begins to use “you”, the phrases are structured in a more how-to, action-based way: “if you wanna find…”. The emphasis then has shifted to an unknown second-person, who is attempting a number of things, whose results or strategies Neves predicts or advises according to his experience. He’s done the work; this imaginary person must, then, be attempting to follow not in the writing of the mathematics, but the understanding and model-building within their own mind.
Let me be less rigorous than Fernando, and just define this as the space of a closed embedded hypersurface, and then you can put a suitable topology…”
Rigour is often spoken about as a defining characteristic of mathematics. From the Wikipedia article on rigour:
Mathematical rigour is often cited as a kind of gold standard for mathematical proof. It has a history traced back to Greek mathematics, in the work of Euclid. This refers to the axiomatic method. During the 19th century, the term ‘rigorous’ began to be used to describe decreasing levels of abstraction when dealing with calculus which eventually became known as analysis. The works of Cauchy added rigour to the older works of Euler and Gauss. The works of Riemann added rigour to the works of Cauchy. The works of Weierstrass added rigour to the works of Riemann, eventually culminating in the arithmetization of analysis. Starting in the 1870s, the term gradually came to be associated with Cantorian set theory.
Mathematical rigour can be defined as amenability to algorithmic proof checking. Indeed, with the aid of computers, it is possible to check proofs mechanically by noting that possible flaws arise from either an incorrect proof or machine errors (which are extremely rare). Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC (see automated theorem proving).
Most mathematical arguments are presented as prototypes of formally rigorous proofs. The reason often cited for this is that completely rigorous proofs, which tend to be longer and more unwieldy, may obscure what is being demonstrated. Steps which are obvious to a human mind may have fairly long formal derivations from the axioms. Under this argument, there is a trade-off between rigour and comprehension. Some argue that the use of formal languages to institute complete mathematical rigour might make theories which are commonly disputed or misinterpreted completely unambiguous by revealing flaws in reasoning. – http://en.wikipedia.org/wiki/Rigour#Mathematics
Though somewhat lacking in references, this reflects my understanding of the issue. Though it’s rather un-rigorous of me to do so, I’ll leave looking up the references for another day.