aligning_normals_ina_low_dimensional_gl_space.pdf |
This paper mostly deals with clarifying some concepts (and tying up some loose ends) from previous gL-related papers.
It's very short....but whether or not it's very sweet will have to be determined by others.
I think this paper represents what I meant in another post when I said "gL-methods/theory can possibly be useful in solving certain problems". The "untangling" of normals here might not be a particularly profound insight to any high level career researcher, but (in my opinion) it's interesting.
I certainly don't think the "gL" will solve any famous open problems (e.g.; Millennium Prize problems). I hope no one ever thought I was saying (or implying) that "the gL project"is going to lead to some giant mathematical breakthrough that's going to solve all the problems. I just think it's an interesting way to compare different types of spaces. It may turn out that, at best, "gL-stuff" is just an interesting pedagogical tool. Or maybe "gL-stuff" will only lead to me getting tarred-and-feathered (with "quack" written on my forehead in permanent marker) at the next symposium/colloquium/seminar I attend. But whatever. It's fun and (imo) interesting anyway.
Enjoy!
(also: over the next few months I'm going to be "tidying up" the posts with previous papers. Writing this paper, and referring back to previous papers, reminded me of how Gog-awful I am at...well..typing. If you want to take embarrassing screen-captures of all of my horrible typos and errors, the next few months will definitely be the time to do it!)
ERRATA:
9/12/18 (that didn't take long, did it?): I feel a little....dubious... about some of these results, especially the F^{3} case. For F^{3}, it may be that you rotate the triangulation by pi (as opposed to 2pi/3), then reflect over plane tangent to the triangulation's most indented point (which is also the point we rotate about). Then the triangulation would have to be translated to its octant's "antipodal octant" (i.e.; the octant diagonally opposite to the original octant).
In both the F^{2} and F^{3} cases, I'm not sure my calculations for differences in angles of laplacians of As & Sq' were totally correct. The methods definitely decrease the angle between any two normals on the given surfaces, but I'm not 100% convinced the difference is 0.
9/17/18: It occurred to me last night (as I was continuing some possibly frivolous side-work regarding a less unwieldy formula for Sq') that, in general, the "gL" papers are mostly written using the algebraic representations of As, Sq', and Ci (and their various higher dimensional analogues) and relating geometric qualities of those algebraic representations. Thus, from what I can tell from books I'm looking at, most of the papers MIGHT (as in MAYBE/POSSIBLY) be better qualified as "algebraic geometry/topology" than the previous qualification(s) of "geometry/topology and/or differential topology" that I had been using.
Honestly, from what I can tell, there's still a lot of overlap into the "geometry/topology and/or differential topology" (as well as the combinatorial analogues listed in the second paper listed on my CV page (listed as reference [1] in the paper above)).
I bring this up because, in terms of "algebraic representation", it IS true that Sq=cos(\theta)+sin(\theta). Note that at \pi/4, cos(\pi/4)+sin(\pi/4)=sqrt(2)= Sq(\pi/4). If one reads "Errata #2" on this site, it's obvious I'm having a real personal crisis in terms of the best way to represent Sq' (basically: I'm bouncing back and forth between "vector-style" and "algebraic-style", but when put in the implied context of the papers I've written... it seems to me that I'm almost exclusively using algebraic-style notation.
And then there's other questions I'm asking myself that are also more algebraic in nature:
1)if Ci(\star) has the integers as its fundamental group, what is the fundamental group of As(\star)?;
2)Can Ci(\star) and As(\star) be used to find a relation between the integers and the hyperbolic integers (which I recently learned are actually a real thing)?.
Also: I've been casually toying with the idea of looking at "gL theory" in the context of representation theory, though I'm still in the "infancy-of-understanding"-stage when it comes to representation theory (which is to say, I'm just starting to get an almost-good heuristic/intuitive non-comprehensive non-rigorous understanding of the field).
10/1/18: For the 2 dimensional case, I'm wondering if it would be better to say "take 2nd derivative of As, then fix point where 2nd derivative of As intersects Ci, and then sweep line created by 2nd derivative from As(p) to Re(As(p))". (Re(As(p)) being the symmetric "twin" of As(p)) It seems more....correct...to go about it that way.