For me, Mathematics has never just been about solving problems I’m given. It’s also about finding new ones. That is why I have always strived to go above and beyond the compulsory syllabus since the beginning. For example, in Year 11, I found out late in the year that I could take the OCR FSMQ Additional Maths Qualification. I challenged myself to learn completely new concepts like kinematics, calculus and optimization in an incredibly limited time. Being in this stressful situation strengthened my ability to quickly grasp new ideas and use them to solve problems that required lateral thinking which I had yet to come across.
At A Level, I took these skills that I had learnt with me to various competitions including the UKMT Senior Team Challenge, Senior Maths Challenge and Senior Kangaroo, in which I was awarded a merit. The time constrained challenges taught me to be much more efficient and, in doing so, find more elegant solutions to problems. For example, in the Senior Kangaroo, there was a question involving two concentric circles with a given chord of the bigger circle being tangential to the smaller circle. When tasked with finding the difference in area between the circles, if I attempted to directly find the radius of either circle, I would have reached a dead end. Instead, I used Pythagoras’s theorem to form an equation identical to the area between the circles but with a factor of pi removed. Within a minute, I calculated that the area was simply the product of pi and the square of half the chord length.
This summer, I partook in the LSE CHOICE summer school and learnt about partial differentiation and its use in optimization with multiple variables. While I did appreciate the application of calculus to real-life economic models, I was much more intrigued by things like the Hessian matrix, which we used to determine the stationary point of a surface. I felt that using it without fully understanding it stunted my ability to further explore multivariable functions. Further investigation revealed to me that a Hessian matrix of a Lagrange function, another tool we used for optimization, is called a bordered Hessian. This allowed me to revisit an earlier optimization problem I attempted with a Lagrange multiplier and determine the nature of the stationary point I found. One part of mathematics that never fails to fascinate me is any field that incorporates complex numbers. When I learned of the Riemann zeta function, I was unknowingly opening myself up to a new branch of mathematics involving complex numbers which could prove that the sum of all natural integers has a finite negative value. This prompted me to begin watching Herbert Gross’s MIT lectures on complex variables to build the stepping stones that might one day help me to fully understand results like this. When watching the lectures, I was impressed by Gross’s use of two different methods to find the derivative of a complex function, initially using first principles and then using known results from derivatives of real functions. This seemed to not only highlight a relationship between the derivatives of these different types of functions but, more importantly, it made me appreciate that complex numbers are in fact just as real as any other number. These seemingly impossible numbers remain consistent within the mathematical model that humans have created, even when more advanced operators are applied.
Hopefully, studying complex analysis at some point at university could even give mean understanding of the Riemann hypothesis, a problem few can comprehend and fewer still can play a part in solving. I believe that, because of the passion and aptitude I have displayed so far for many facets of maths, I would be a perfect candidate to study for a degree in Mathematics.