This paper seeks to justify the hand-wavy claim made in previous papers that gL can be used to solve the sharp corner problem. Yeah…that’s… a doozy, and I was really hesitant about posting this paper. But… if you feel something is correct, sometimes you just have to take the risk of being wrong, yeah?
I tried my best to really emphasize that the result is completely dependent on gL. In fact, the word “conditional” shows up 9 times in the document (including the title), and it’s used in the phrase “conditional solution” 8 of those 9 times. I also really tried to emphasize that it’s not a perfect solution. The final theorem is definitely NOT an IFF statement. It’s very much a one-way argument. And there is definitely emphasis on the fact that there is a jump continuity when taking the limit of the derivative in the negative direction, even as we’re able to define a derivative at every point, including zero, for Sq’ under the gL construction.
I tried to really refine the idea of gL itself here, and I think the result is a more coherent idea of what gL is and how it’s built.
I also tried to be less hand-wavy in general. I hope I succeeded there. I definitely feel that it’s LESS handwavy than previous papers, but as to whether or not it’s “hand-wave-free”…I’m not 100% sure about that. The presentation of Sq’ is definitely less hand-wavy here, anyway.
By and large, I do feel this one looks and reads more like a real math paper than previous papers. There are actually definitions, propositions, and a theorem. AND there’s proofs for the propositions and the theorem. A great departure from previous papers, which I’ll admit read more like notes to prepare for writing a rough draft than actual papers.
One concern I have is whether or not the 1st proposition even needs to be a proposition, or if it could just be a definition. I chose to approach it as a proposition because I felt like it was worth proving it and not just stating it as fact.
As with previous papers, all of the references from previous papers are applicable here, but I only give the previous papers as references.
Also:
- A special thanks/acknowledgement to Prof. Tevian Dray for: a) offering some really solid critiques of my technical writing style in my homeworks for MTH 434 and MTH 435 (the 2-term differential geometry sequence here at OSU), which I’ve applied here, and for b) introducing the function I’ve adopted here for Sq’ in MTH 435 class discussions. (Note: Prof. Dray did not directly advise or help with the actual content of this paper. Any quackyness in the paper is attributable to me and only to me..)
- An anecdote: The writing process for this paper was… interesting. As I was doing LaTeX formatting/editing in Overleaf (shameless plug for my favourite TeX editor!), “my” internet decided to go out. And then it never came back…except for in my kitchen, by the window next to my stove. This paper was entirely typed and edited on benches on OSU’s campus and in a chair in my kitchen sitting next to a window while leaning against a stove with my feet propped up on my kitchen counter. Basically, the typing/editing process was Level-10 Ghetto-ness.
- Readers of past gL papers will notice I avoid calling gL itself a manifold. I was (sort of) in error calling it a manifold in previous papers. I think any mathematician would agree that, as is, gL does not pass the manifold test UNLESS we preform surgery and induce a connected sum in an infinitely small (but not emptry) set about the intersection points. In general, I'm trying to make gL a structure/construction that's adaptable to different ideas. Here, I didn't want to lose the notion of single-point intersection, since it's crucial to the result. So gL is most definitely NOT a manifold in the paper below. In other papers (written and to-come), I may wish to make gL a manifold, and will/would thus have to use proper surgery methods to be able to call gL an actual manifold. Anyway, I hope that clears up any ambiguity and/or confusion with previous papers and the use of "manifold" in reference to gL.
ERRATA:
1) Limiting to the unit case for Ci, As, and Sq' is totally unnecessary. Every result in this paper can be extended to generalize to the appropriate functions on any radius r in any dimension that the n-sphere exists in geometrically.
2) As indicated above, the results are obviously limited to dimensions where the n-sphere can exist.
3) Stylistic Quibbles and Quibbles/Uncertainties/Musings about gL in general:
a) While I mentioned this was "less handwavy than previous papers" above, I still feel like I'm writing too loose. I want formalism and a more defined symbolic representation. I feel like I'm still leaving too much left unsaid, when I should be striving for formalism.
b) Related to a), I still feel like I'm being too broad with the subject matter. I don't feel like I'm being specific enough with the spaces I'm working in, for example. Mostly, I've been showing Euclidean space, and then making vague statements about hyperbolic spaces. I think my next paper should be a retread (w/ improvements) of old material/papers, but for the hyperbolic case. I definitely want to refine/correct the "aligning normals" paper, and it would be nice to add the hyperbolic case.
c)Related to a) and b), these papers are way too vague in terms of having any natural home in any specific subject or intersection of subjects. There's no real context to anything. Some times it's all local qualities, sometimes it's all global qualities, sometimes it's both. Sometimes it's vaguely topology, sometimes it's vaguely geometry, sometimes it looks like it could be algebraic or analytic, sometimes it looks like gobbledeeguck. It's, again, too broad. For a while I've been feeling like Symplectic/Contact Geometry/Topology might be a good place to start, in terms of giving gL a "base point" to explore from. I believe that I previously stated in another post (maybe an errata?) that, as far as I could tell, all of the component manifolds for gL are Khaler, which would allow them to be studied, at least individually, in the context of symplectic geometry/topology. Studying gL as a symplectic manifold would (as mentioned earlier in the post) require the proper methods from the theory of surgery on manifolds (certainly the idea of connected sums is relevant). I'm also working under the assumption that the connected sum of Khaler manifolds is, itself, Khaler, which could most possibly be true or not. I'm not 100% sure. Obviously, the component manifolds themselves are pretty uninteresting (in any subject, not just symplectic or contact geometry/topology). I'm kind of hoping that the composition gL WOULD be interesting, though. Even certain odd n cases could be interesting in terms of contact geometry/topology. I don't think it's insignificant that this whole gL thing was inspired by the geometry of optics, which is an aspect I really want to get back to in future work with gL. I've really glossed over, if not outright ignored, the nitty-gritty of the optics aspect.
At the same time, the gL thing started as being intentionally general. It was meant to be as universal as possible. So maybe it's not so much that I need a certain "home base" subject so much as... I just need to start applying gL to SOMETHING to start actually using it (or at least, seeing if it's actually usable....or just boring and trivial). And symplectic geometry/topology seems, again, like a natural landing spot (especially given it's reach into so many other subjects like Representation Theory, Algebraic Geometry, and so on...with my more general interests of [differential] geometry and topology being implied, obviously).
I should note that I do, in fact, know that the sphere cannot be symplectic. But, from what I can tell, the hyperbolic octahedron can be symplectic. So a lot of the ideas behind aligning normals of the the hyperbolic octahedron and the sphere can...maybe?... be seen as a way to create a relation that allows us to view the hyperbolic octahedron as a symplectic sphere. (This is why I need a good symplectic mentor... so they can shoot down any crackpottery before it gets too out of hand).
(Update 11 Nov: Ok...so the 2-sphere IS symplectic, but the 2n-sphere for any n>1 is NOT symplectic....BUT all odd dimensional spheres ARE contact. Writing this update more for my sake than anything else.)
d) I've been really shy about introducing discussion on applicable differential equations in all of the gL(\star) papers. I've been intentionally vague with regards to the differential equations because (as with a lot of different aspects of these papers) I don't feel familiar enough with certain material. My knowledge of differential equations on manifolds is...ok.. I did well in my Intro to ODE's course (MTH 256 here at OSU... I got an A-, which is pretty ok), but aside from touching on "ODE's on manifold" in differential geometry, I haven't had much exposure with that specific material. It's something I want to address/show in future papers, and I'm definitely working on that aspect.
4)I'm trying to think if this all boils down to just, intuitively, a union of Axiom of Choice and showing homotopy equivalence under a specific topology? I'm pretty sure this construction of gL presented in this paper (and the resullt of this paper) qualify as separated spaces. With the conditions given, this should make the derivative map of Sq' in the positive direction Haussdorff. I'm almost positive that in the negative direction the derivative map is NOT Hausdorff. (This is another example of how I use these papers to practice and synthesize things I've learned, ps).