Cryptography & Number Theory
Outside computer science, I am also deeply curious with mathematics as a whole. Having self-taught both Calculus I & II in high-school, I’ve grown an interest in the field of pure math, and through my undergrad program, have discovered an interest in number theory.
As of now, I am focusing on my career, though I do plan to eventually comb my way through abstract algebra and number theory as a means of eventually learning the mathematics behind modern cryptography.
This can help gauge a better understanding of how encryption, decryption, and hashing ultimately work, their pros and cons, and how the scene shifts through the eventual, and inevitable, rise of quantum computing as the cybersecurity field moves toward post-quantum cryptography.
Systems Theory & Category Theory
Through my experiences, I also found myself drawn into thinking about systems and the relationships between components, rather than through the components themselves.
The mathematical version of this is what’s known as category theory; I plan to learn category theory at some point, though I’m unsure as to when. Given my interests into diving deep into the underlying frameworks and engines that allow things to run. diving into category theory seems as the ultimate step into truly learning what mathematics really is.
Mathematical Extremes & Infinity
Systems aside, I have also found myself drawn toward mathematical extremes; whether it be the recursive nature of Cantor’s infinities in set theory, or unfathomable finite numbers, or abstract higher dimensional geometry. Even if I don’t fully understand the mathematics behind them, I appreciate them through an artistic lens.
During 10th grade is when I started independently grappling with the idea of infinity. I wondered to myself: how can the number of integers possibly equal the same number of real numbers? Given that the amount of numbers between 0 and 1 is infinite, it seemed wrong to assume that the number of integers is the same, given that there seemingly infinite infinities in-between each pair of integers. I was sure that the number of real numbers was bigger than the number of integers, even if they’re both technically infinite.
This led to me discovering about Cantor and set theory, and most notably, Cantor’s hierarchy of infinites; from his proofs that permanently scratched that itch in my head regarding the real numbers to the inevitable expansion of infinites seemed completely mind-boggling, yet beautiful at the same time, in an artistic and humbling way.
- This newfound nature of my infinity that I discovered inevitably found itself into my worldview.
Higher Dimensional Geometry & Polychora
Similarly, over the years I have wondered about just far geometry can go. I have also been interested in the idea of higher dimensions mathematically - experimenting with multiple axes (up to six dimensions) by putting new ones in between two pre-established axes.
This led me down a rabbit hole over the years, and I eventually grew an interest and love for polychora, or more specifically, nonconvex uniform polychora. While I will not pretend I understand it all fully, I view it similarly to how I view infinity; i.e. I primarily admire them through an artistic lens in the sense that they are genuinely impossible to fully comprehend, yet we can still, in part, admire them and discuss them.
- This idea of “overt simplifications” as I call it also find themselves in my worldview.
Googology
Within that same realm, I also am pretty interested in googology, and seeing the limits of numerical definitions. From Graham’s Number to TREE(3), the Busy Beaver Function, and ultimately, Rayo’s Number, it’s completely mind boggling how enormous numbers can get.
- I plan to construct a mathematically viable “large” number to compete with the likes of these numbers.