Why do we care about Topology? The topological Tverberg's Theorem

This is a continuation from a 3Blue1Brown video with the same title, an excellent video you should definitely check out!

Earlier this summer, I had the incredible opportunity to participate in SUMaC, an opportunity that allowed me a glimpse of the wonder of pure math, specifically Algebraic Topology, which is essentially a study of how shapes can morph and deform, and when two shapes are equal (formally, a homeomorphism between topological spaces). Think of the quintessential topologist's dilemma, always mistaking their coffee cups for donuts (both homeomorphic to a torus). 3b1b showed a fantastic connection between the inscribed rectangle problem and the weird topology of the Klein Bottle. But it turns out these connections also exist in discrete geometry. One amazing example (also the topic of my SUMaC presentation) is the Topological Tverberg's theorem.

Wikipedia can probably do a better job at explaining the theorem than I can (link: https://en.wikipedia.org/wiki/Tverberg%27s_theorem). Specifically, it says that if we take (d+1)(r-1) + 1 points in d-dimensional space, then we can make r groups of points where their convex hulls all intersect. While the discrete geometry proof works with affine transformations and a bunch more geometric mechanics, there is a (somewhat) natural extension to this question.

So far, we've been working with convex hulls of finite points, which essentially means polygons in euclidean space. But what if we weren't using euclidean space, and instead of polygons and straight lines, we wanted arbitrary curves. Turns out this result is still true!

More formally, it deals with continuous maps from a d+1-dimensional simplex (basically tetrahedrons, but generalised to higher dimensions) to d-dimensional euclidean space (the key part is it's one dimension less), and claims that there is a point such that the images of two distinct faces on the original simplex include the same point. This means you can pick any function! Absolutely any function you want, as long as, if you move across the simplex, there are no sudden jumps in space (i.e. it is continuous).

This also works for q(d+1)+1 points, and q faces map together. While I don't have time or space to cover the full proof (you should check it out if you haven't yet), the general idea is to label each of the faces with a number, then use the generalised Borsuk Ulam theorem (watch the 3b1b video!!) to arrive at this answer, then continue to q > 2. I know this is a significantly superficial explanation, but hopefully it piqued your curiosity to go see the full thing!

Here's a nice slideshow on the history of the theorem: https://faculty.sites.iastate.edu/zerbib/files/inline-files/TopTverberg.pdf