Apologies if this isn't the right place to enquire about this; I'm really looking for potential connections to what I've been doing as opposed to having a maths question answered. With that being said, I would heavily appreciate tips and advice as to potential techniques as well as connections.

Background: I'm currently a first year university student (about three-quarters of the way through first year) so I'm not well-versed in higher topics in mathematics. Currently doing multivariable calculus and differential equations, and have had a rudimentary introduction to linear algebra (and plans to do real analysis and linear algebra next year). With that being said, the whole point of me asking here is to get answers as to where I can go in relation to some of the stuff I've been doing.

I have a bit of an unnatural obsession with piecewise objects in general, and I've been told that some of the stuff I do has the potential to dive into various areas like topology, etc. So I run a site plainly called Piecewise where I really sort of explore and post some of the stuff to do with piecewise.

More or less - I'm looking for places to go with some of the stuff I have been (and have not been) doing with piecewise objects. Even more specifically, I base a lot of the stuff I do on the foundation that the cases within piecewise objects aren't independent of one another; i.e. operations you perform on one case you perform on others in order to relate each case to one another.

Current work: The above idea has yielded some interesting derivations, including the ability to interpolate a set of points as polynomials (with quite-flexible ideas), which was entirely accidental and had been a goal, that is, turning a set of points into a polynomial without solving a system of equations. That was one of my goals a couple of years ago during VCE which ended up going nowhere till more recently, when I was told that what I had derived appeared to be a lagrange-form polynomial; a polynomial derived as a consequence of lagrange interpolation. Later, I also rederived interpolation as newton-form polynomials using a similar method.

Other things I've been able to do have largely been geometric and graphical in nature; deriving a continuous batman equation/relation from the original, equations of hollow hypercubes in n-dimensions, a formula for the sticking together of several functions along a given axis (I keep hearing things about the gluing lemma in this regard?) which can be extended into higher dimensions on surfaces, but takes on a far more ugly and restrictive formula. Obviously a lot of these things are fairly rudimentary in their approach and nature.

Problem/Question: Naturally, I'm absolutely curious about where I can go with these ideas and potential connections to higher-level mathematics, not necessarily geometric in nature (although it seems to be a recurring theme). It furthermore seems to be an issue that with my peers, the line of thinking I'm on takes time to understand, so I have found it difficult to communicate my ideas a lot of the time - but have had occasional assistance.

Specifically, what areas of maths might be the most relevant to what I'm doing, and, what potential subtopics could have some potential relevance to these piecewise objects, if any? Additionally, from what I can see, no one else has really approached piecewise in this way before that I can find, which has led my friends and I to a few hypotheses:

  1. Exploring piecewise in this way is largely redundant and ultimately leads to very little that generalises to higher mathematics.
  2. It has been explored, and all exploration that has been done has ultimately been integrated into other things (wherein perhaps piecewise was used as a scaffolding that was later removed).
  3. It genuinely hasn't been explored before (and imho the least likely option).

Naturally, I am also curious as to (1) whether I'm expending my time that on something will yield little or is 'useless', and (2) did I miss something? Has this been done before?

Thanks in advance - and apologies if this isn't specific enough (also had no idea what to tag this post with).

  • $\begingroup$ Would also appreciate knowing as to why I've been downvoted too so I can improve my post. $\endgroup$
    – Ally
    Sep 9 '21 at 5:59
  • $\begingroup$ Personal advice is generally considered off-topic here, and in any event is generally better asked of someone who is familiar with your mathematical background, e.g., a lecturer at your university. $\endgroup$ Sep 9 '21 at 6:55
  • $\begingroup$ A suggestion of one of the lecturers I did in fact contact was to post here specifically about the potential connections to the work here, which is what I'm asking about, and less about 'personal advice'. As it stands, there aren't lecturers who are familiar with my background, given, for one, that I am a first year student and don't actually have background, and for another, lecturers who I have actually contacted are busy. I see no alternative place to post this, but suggestions are welcome. Thanks for taking the time to read my question, much appreciated. $\endgroup$
    – Ally
    Sep 9 '21 at 7:00
  • 1
    $\begingroup$ Have you looked at splines at all? mathworld.wolfram.com/CubicSpline.html might be a good place to start. $\endgroup$ Sep 10 '21 at 6:21
  • $\begingroup$ Hey! Yeah, and I've been doing some parametric curves - in fact as part of interpolation - where splines in particular have been inspiration. Especially in higher dimensions, which has been quite tricky to model (one method will give a surface, another gives a space curve). So have been working out the relationship between parameterising everything and whatnot. Thanks a heap for the link, will shift focus to splines directly soon I think ^^ :) $\endgroup$
    – Ally
    Sep 10 '21 at 9:56

In topology there is the notion of a simplicial complex which can be used to approximate smooth manifolds via triangulation. This then can be used to effectively compute homology groups, whose abstract definition is rarely viable to do computations. So if you are interested in shapes and geometry, I would suggest to take a topology class in a maybe a year or two (depending on your speed of study). Though you will need a solid base on linear algebra and group theory going into algebraic topology.

I also took the freedom to remove the algebraic-geometry tag from your question, because Algebraic Geometry is actually an area of math, where we do rarely deal with triangulations.


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