[WP2] The Process of Visualization in Abstract Research
WP1 reflected more on the “what is this connection between two seemingly unrelated fields”, and was very broad in terms of just describing the connection itself, and some of its larger results, and was general, as it was relating to the entire field of math. WP2 is a more specific deep dive into “how this specific connection is developed, and what we use it for”, and provides a proof of concept itself, with a more specific mathematical concept in use, and its relation to a more specific part of a garden- trees. This also, to some extent, explains my current research as a learning mathematician in relation to common objects, why we need to make a connection of some sort because of the abstraction, and how one would go about doing this. We further show another large unsolved problem in mathematics, and how the common man may think of it and present us with ideas of new approaches of solving it.
This is what math research looks like when it’s put in an actual paper, this for example actual research I’m working on right now. Pretty scary, right? In an article I wrote a little while ago titled Math, Gardens and the Power of Visualization in the Pursuit of Knowledge, I spoke about how we can connect these equations to gardens, and have some form of viewing them in a less abstract way to make them easier to work with, and more accessible, which makes it easier to produce more developments in the field. This is the main reason I’m writing this essay.
The reason I like to treat these mathematical objects like real objects you can feel and touch is the idea of being able to ‘see’ them, and find unique ways in which they work, which may be similar to how things we see everyday work too. It’s a good way to shorten the amount of time spent thinking about a problem, and can help create tricks to solve problems quicker, like, would you rather try attempting to break down each of these equations into their respective parts and try solving each individually and combine them again to find a solution, or try to figure out how this can be seen as having a physical meaning, and try solving it visually? This is a 3Blue1Brown video about solving a relationship between pi and the sum of inverse squares from 1 to infinity, using lighthouses and distances, which I would strongly recommend you to watch. Speaking of relationships, my paper aims to find correspondences, which are essentially bridges between different objects- I am currently working on constructing these objects called pseudo cusp forms, using a correspondence they have to other functions called theta series, and also trying to construct a bridge myself, between pseudo cusp forms and eigenfunctions. As it would require a lot of background knowledge to understand exactly what this means, you can think of it as me building an island around a single tree attached to a bridge connecting it to a city, and also building a different bridge from that island to a much bigger city. As a mathematician named Hejhal pointed out a long time ago, there is also a foggy country in the distance, which would take a lot of effort to reach, but could unlock a lot more potential if solved. This is a possible object that may be used to attempt to solve the most famous math problem right now- the Riemann Hypothesis.
Notice that at the very center of every connection is a tree, and the cities are chosen carefully. We choose a tree to represent a pseudo cusp form, as there are special kinds of each corresponding to each city- branches with specific properties, from which these bridges are built if you may, and it is part of a whole group of trees on the island called automorphic forms, which is a group of functions studied by number theorists, which are considered ‘the fifth elementary operation of math’, along with addition, subtraction, multiplication and division. The way we make these connections is by brainstorming, breaking the problem up into different ways, and finding things that possess similar properties, and then seeing if there can be some group of objects possessing all of these, such as my city and island and so on. In this case, since we are talking about correspondences, we use the analogy of bridges- which also need material and a means to construct, in this case, my research needs a lot of other proofs of certain theorems and so on for me to develop these connections, and as we refer to eigenfunctions and theta series as cities, this is due to them currently being used to develop technology. We refer to the solution of the Riemann hypothesis as a completely different country, as once it is solved, we will have access to things we have never seen before, such as its applications in cryptography, which would essentially lead to unhackability.
We also look at an application of math in physics, and physics IS an explanation of how the world around us works, literally, and we look at string theory, which is supposed to explain how every kind of particle interaction works, and is, once again, explained through a hypothetical real everyday object like a string, although everything about it is abstract. It essentially states that every particle can be defined using some kind of imaginary string, whether it’s curled or not, open or closed, and using ‘spinors’ which essentially are different ways the string itself can curl. This is very close to our explanation of everything in physics, and extremely important- can be explained by simply a visualization. There is a huge reason that physics is a lot more ‘newsworthy’ than math, and that’s because the results can be SEEN and explained to people in an easy way, and my aim is to get more mathematicians to be able to do something like this. For example, the Riemann Hypothesis, which is implicitly related to my research right now (which is in a way relating to proposing it in terms of Pseudo Cusp Forms), is about the location of the zeroes of the zeta function lying on a line with Real part ½, which is just terrifying to explain to someone and establish its importance, which is something I one day hope to tackle, but hopefully this provides some explanation of the importance of visualization and how it is made.
Sources Used:
[1] XUE, Hui, and Naveen Prabhath. “ACTION OF E2 ON HECKE EIGENFORMS.” Kyushu Journal of Mathematics 77.2 (2023): 401–412.
[2] D. Hejhal, Some observations concerning eigenvalues of the Laplacian and Dirichlet L series, in H. Halberstam and C. Hooley (Eds.), Recent Progress in Analytic Number Theory . Academic, London, 1981, 95–110.
[3] 3Blue1Brown. “3Blue1Brown.” 3Blue1Brown, 2019, www.3blue1brown.com/.
[4] Edwards, H. M. (1974), Riemann’s Zeta Function, New York: Academic Press, ISBN 0–12–232750–0, Zbl 0315.10035
[5] ScienceClic English. “String Theory.” Www.youtube.com, 2 July 2021, www.youtube.com/watch?v=n7cOlBxtKSo.
