This errata alert concerns papers 1 through 3.
The functions with respect to a variable "t" are (in fact!!) parametric functions, while functions with respect to theta are polar. I realized I kind of mixed up notations in my text(s) of papers 1 through 3. I believe some equations in the papers and in lecture notes were written with the radius as a function of variable t, when it was meant to be shown that the radius is a function of theta, but dual to an expression in variable "t".
OOPS!!!
I'd also like to add that, as it stands and to the best of my understanding, the concept of a Geometric Lens as an analytical tool is not related to the notions of lens spaces or fake lens spaces. My understanding is based on what little I've read on both (notably, "A Short Survey of Lens Spaces" by M. Watkin [1989/1990) and "Surgery On Compact Manifolds 2nd Edition" by CTC Wall [1st ed, 1970, 2nd ed. 1999]). While I'm sure there's intersection with both Lens Spaces and Fake Lens Spaces and Geometric Lens...Theory/Methodology...I do not see any indication that these ideas are (completely) dual in any way.
I do think it might be fun to look for applications of LS/FLS to gL-type analysis, but that undertaking would be done well in the future.
Also, I HAVE (eventually, in my own time, obviously) reached the realization that I'm essentially working with Lie Groups. as these are definitely known differentiable manifolds which have algebraic meaning (which I wasn't really going for in that 4th paper) in the sense that we can consider the determinant under the real (or complex) general linear group as a type of Lie Group algebra.
I'd also like to formally apologize for my HORRIBLE editing skills when typing final drafts of papers. This is especially true of paper 4's conclusion. I have a really odd habit of typing everything in LaTeX code in a normal document file like word so that I can be more comfortable editing, and then I copy and paste it into the tex editor and edit the coding errors. Sometimes I miss typing errors in the word document, and then I get so focused on the coding that I forget to double check my typing.
In the case of paper 4's conclusion, I forgot to completely edit out a note I wrote to myself about circle/sphere packing while I was still toying w/ the idea of whether or not it was too half-baked to include. So essentially, the conclusion reads as some sort of run on sentence formed my my thoughts and my actual output. DOUBLE OOPS!
2/26-ish/2017: I wanted to add a clarification in regards to the notion of the 1-manifolds being surfaces. What I mean to say when I call them (the circle, the aster, and the square rotated by pi/4) surfaces is that, with respect to their 3 dimensional analogues (sphere, hyperbolic octahedron, octohedron respectively) the 1-manifolds and their scalars are representative of a partition of the 3-dimensional surface. For example, the unit circle is a (sub)surface in that it has a direct representation as any geodesic on the sphere. So the circle is a "surface" in that it is a component of the large spherical surface. Without the circle, the sphere is no longer the same completely connected, closed surface we know it to be.
I didn't feel like I made that clarification explicit in any of the papers where I refer to 1-manifolds as surfaces (papers 2, 3 and 4, and also lecture notes for talks based on those particular papers). So there you go.
3/1/2017: I'd like to add, more as a "caveat" than an "error correction", that I realize I have a more.... intuitive approach to mathematics. Sometimes (ok, most times) in my papers I choose being "wordy" over being more rigorous in my symbolic language. It's something I'm definitely working on. Most times I read through my papers after I post them and my biggest complaint with my own work (aside from typos and poor editing) is that I'm writing out things that could be more succinctly (and accurately) expressed in symbolic terms.
I feel my work suffers a bit, also, by not having a more standard mathematically logical approach ( ie; having definitions, lemmas, etc. listed as "supportive statements" to claims I'm making). I admit that I leave a lot to implication. Sometimes I'm also unsure of just how much I need to define in a given paper (eg; in working with heat kernels on a manifold/surface, do I need to spend a whole section or even chapter setting up the Theorems and Lemmas and such that lead to the definition, or is it safe to just give the working, standard definition of a heat kernel and assume my audience knows all of the "mathematical backstory" I'm talking about?). Part of this is simply due to my inexperience (I'm only just now completing Advanced Calculus, for instance) and another part of the problem is, at least to a certain extent, my lack of outside/extracurricular guidance.. I think, to a certain extent, my lack of guidance allows me freedom, since I end up with a rather large amount of... creative independence., but it also hinders me in that I also end up having a lot of loose ends that don't get tied up because I don't have outside critique and the occasional nudge towards particular logical consequences I either overlooked or was utterly unaware of. In a way, my current "test" is to find a way to retain my creativity/independence while also tempering it with others' opinions and (constructive) criticism and suggestions.