Why things should break [Part I]
In any decent course of Engineering, there generally exists a minor topic either presented as a marked course, or an overlay over other papers across semesters, that needs to be studied more. I am, of course, talking about Structures & Stability. Wasn't it obvious from this note's title? It should be. Because structures correspond to stability, which determines whether something breaks or doesn't. In engineering (and life, in general), it is critical to continuously evaluate things on these metrics. How do you know your finances are going to hold? How do you sleep comfortably with utmost faith in your health? What if push comes to shove? What if...?
That single question can ruin your peace of mind if you don't know how to tackle it. And that's where we enter with our secret weapon: Catastrophe Theory.
The Idea
In life, things are either smooth or rough. There's nothing in between. Which doesn't say much about reality, but rather our own vantage point. From far off, a blanket looks smooth. Come closer and wrinkles pop up, fibers stand out like grandpa's bald head and smoothness is lost. This texture is coarse-grained. And it is the same across a number of fields. Population dynamics, String theory, Deep Learning, Ecology, and so on. Anything that has to do with data will inevitably require us to analyze the "graining" of its texture. This is an interesting topic on the geometry of data that we shall delve into at some point, but turn your gaze to our blanket now. The coarse graining reveals the structure that binds every strand together into a coherent whole. The smoothness from afar reveals just the information about its stability. Both are important, and interlinked. And it depends on what you want to see, when you analyze the graining of reality. So how do we predict when things might fail?
We don't. Instead, we make hypotheses. We take data and try to build correlations and piece together a plan that might explain the apparent failure of things. One of these hypotheses that has been mathematically shaped into a theory is Catastrophe Theory.
Singularities
Everything we observe is the solution to some differential equation. Sure, there might be systems of connected equations, and most of them are nonlinear, which simply means that one particular thing might be the solution to many different systems, or a number of solutions might exist for one system, or (and this is important) you cannot really trace a solution to its equation. Regardless, every solution space unravels some part of reality, and in every solution space there exists a trace of the inexplicable.
When an equation fails to describe a system, not because of any semantic error, but rather failure of "scope", it might end up revealing a singularity. In mathematical language, these structures are called "poles", "zeroes" and a couple of other names. And there are differences in their natures as well. However, what is common, is that at each of these points the equation fails. The entire mathematical structure breaks down, and we need some assembly to actually figure out the structure of the singularity. Imagine a sort of mathematical black hole (pardon me purists) where information vanishes and must be retrieved through sophisticated mathematical sorcery.
And this sorcery can actually be learned. In what shall be a series of posts, I shall chalk out my learning about this field of mathematics called Catastrophe Theory, and share my thoughts and ideas as they emerge. This is the first, and definitely not the last of the series.