I've been working on this whole "geometric lens" thing for almost a decade now, and I've been sort of searching for validation that it's not a totally worthless endeavour.
The other night, I stumbled upon a some what recent (by mathematical standards, anyway) paper by Vladimir Arnol'd called "Astroidial Geometry of Hypocycloids and the Hessian Topology of Hyperbolic Polynomials", and it was like.... maybe not a total validation of my work, necessarily, but a feeling like what I've been doing isn't total crackpot-ery.
It took me a while to find and acquire the actual paper, but last night I was finally able to download a copy off of IOP Science. I've only started skimming it, but wow... Not only does the paper give me new life (and confidence) in terms of desire to push on with the "geometric lens" stuff, it also gives a bridge from that work to the Symplectic/Contact Geometrical world and the Algebraic Geometric world.... all of which are major interests for me. I had an idea of intersection of my gL work with Algebraic Geometry (see my most recent paper), but this paper by Arnol'd definitely shows me there's higher peaks I can climb to if I put the work/practice in.
I'm definitely about to hunker down and go over this (70+ page) paper, with some other books that will help me fill in some gaps, and start a new paper. This is the kind of thing that motivates me to learn the fundamentals; finding an interesting subject or paper that really sparks my curiosity. In this case, the fact that there's significant overlap between my pet project and my other interests makes this find even more so a motivator.
But it's good to feel not-crazy, it's good to have the validation that a mathematician of Prof. Arnol'ds status was doing similar (if not better, more rigourous, and well articulated) work, and it's good to have that extra wind beneath my wings at a time when I was feeling really frustrated.
Finding (and attaining) the paper was just such an exciting and pivotal moment for me, and I had to share.
My next paper is definitely being dedicated in Prof. Arnol'ds memory. I can't thank him in person, but hopefully I can make my work a memorial to him, and to have my work also be worthy of his work.
Actually, I might take this moment to discuss the...I guess journey?...that this whole geometric lens thing has gone through. It's definitely a project that's seen it's fair share of (very necessary) edits and corrections. I started it with little more than a sophomore level vector calc background and a lot of self-study. But self-study only takes you so far. There's always going to be gaps of understanding at any level of math, from student to professional. No one can know everything. But I caught the research bug after the first (mostly expository) paper I wrote, and I ended up trying to consume everything from basic linear algebra to current research without having complete understandings of what I was consuming. I think one of the best examples of this is my use of/appeal to algebraic fields in early papers. I had a limited understanding of what a field was (at the time, I understood a field to be a natural, integer, rational, real, or complex number system with operations. Then I learned that those weren't the only types of fields. And I still haven't gone back to edit for (or outright correct) that particular mistake (making the claim that a result or two "held over ALL fields", which was only definitely correct with my incomplete understanding of a field...whether or not my claim is correct over ALL fields, or only over some fields is something I am not sure of as of this moment).
I've definitely had some bonehead errors, too. I think I've talked about how I've left them up because (a) it's a good reminder that dumb mistakes happen if you don't pay attention, and (b) to always make sure I keep myself humble. Nothing keeps you more humble than looking back at notes from a talk you gave and realizing you wrote cos(x)=cos(t) and sin(y)=sin(t) as a change of variables on a white board in front of a room full of mathematicians.
Still, despite the fact that I was using the language of mathematical infancy to describe things that required advanced mathematical language, I feel I have come away with some useful stuff. For example, I still stand by the "aligning of normals" methods for gL1 & gL2, even if the results themselves need some tweaking*. And there's a lot of conjectures I made that I stand by. For example, I mentioned in at least one paper (or the post it appeared in, at least) about the importance of the fact that As^n contained cusps, and desire to look into hypocycloids and the super-ellipse. The paper from Arnol'd was a validation of my intuition. Now it's a matter of moving beyond intuition.
*It's especially true in the gL^2 case in the "aligning normals" paper. Aside from the fact that gL^2 is more of a case of aligning binormals, I'm not 100% sure the value on my rotation transformation to align the binormals is correct. I also think it would be interesting to explore the tangent and fibre bundles of gL^2. since the tangent bundle of each element of gL^2 (via its component 2-manifods) a 4-manifold. The "unknoting" given by the rotation transformation would give a nice projection mapping. It's something I'm working on now, between (re)studying basics and trying to keep the school thing a reality.