Mathematics Personal Statementfor Oxford, Cambridge & Imperial

Reading about the gap in Andrew Wiles's proof of Fermat's Last Theorem before the corrected version appeared in 1995 changed the way I thought about mathematics. Until then, much of school maths had felt like a set of techniques that either worked or did not. Wiles's proof made me notice a stricter standard: an answer is only secure when every step can withstand scrutiny. What stayed with me was not just the theorem itself, but the fact that a claim about whole numbers needed ideas far removed from the arithmetic that first produced it. That made number theory feel less like a collection of puzzles and more like a subject where simple questions can force genuinely new mathematics. At university I want to study mathematics in a way that keeps those questions open rather than smoothing them away. I am most drawn to number theory, analysis and the theory behind numerical methods because they all test the distance between getting an answer and understanding why it deserves to be trusted.

That question became more precise in Further Mathematics when I studied numerical methods. Newton-Raphson first looked almost too efficient: one tangent could turn an awkward equation into a sequence of much better approximations. What interested me more was when that efficiency failed. A poor starting value, or a function whose behaviour made the tangent misleading, could send the iteration away from the root rather than towards it. Comparing this with bisection made the trade-off clearer. Bisection is slower, but if the interval really brackets a root then each step preserves that guarantee. I liked that the comparison was not just about speed. It made me think harder about what mathematicians actually want from a method: elegance, reliability, or some balance between them. I explored the same tension between calculation and justification in my EPQ on when iterative methods for solving equations are reliable. I wrote a short Python program to apply Newton-Raphson and bisection to the same equations from different starting values, then used Desmos to plot the iterations and compare what the algebra suggested with what the graphs showed. The most useful part of the project was not confirming that Newton-Raphson is often faster. It was finding cases where it looked convincing for a few steps and then became unstable, or where convergence depended much more on the initial value than textbook examples had led me to expect. I could often describe the pattern before I could explain it properly, which forced me to separate noticing behaviour from proving why it occurred. That is where analysis began to feel important to me rather than just difficult.