When the results of the final exams were announced at my school, it turned out that I had won the prize for the best Abitur exam in mathematics, which covered a trip to SaarbrĂźcken, where the annual convention of the German Mathematical Society took place this year. Many internationally renowned mathematicians were invited to that convention to give speeches on recent discoveries and theories in their respective areas of expertise. The topics very highly diverse, including the computation of Navier-Stokes-Fokker-Planck systems, image processing, machine learning and arbitrage.
I've had always known Socrates' statement, "I know that I know nothing", to be true; however, it did not quite prepare me for the complexity of the speeches: while the lectures offered a vague outline of the solutions to the challenges they dealt with to the uninitiated, attempting to truly grasp the involved theories demanded a lot of work. The fact that even professors who happened to specialize in different realms admitted to not understanding everything offered little solace.
Nonetheless, the humbling experience did not put me off my interest in mathematics. On the contrary, I experienced a huge diversity of highly interesting areas to potentially specialize in, and met some fellow aspiring mathematicians, who, like me, maintained their tenacity and ambitions rather than be intimidated by the sheer extent and complexity of mathematics.
The convention has demonstrated very palpably the ever-present trade-off between depth and breadth of knowledge. That trade-off does not mean, however, that one should sacrifice one for the other. On the contrary, a broad range of interests means that there are many enticing areas to gain deep knowledge in. Possessing both deep and broad knowledge is highly helpful in finding interdisciplinary relations, and thus eases the advancement of one discipline by utilizing experience gained from another.
I love this, but I think it is really important to connect this to what you're going to do in college--Stanford has wonderful interdisciplinary programs, and you should make it clear that you know that and that's a big part of why you want to go there AND why you're a great fit for the school. You can tighten the story a bit so you have room for some of this. Good luck!
