<p>Hi there,</p>
<p>I’m applying to grad school for mathematics, but am a bit unsure about my personal statement. Could anyone give me some feedback? This is a first draft. Thank you! (Also is this the right forum for this? It seemed to be the only grad school forum.)</p>
<pre><code>I love the paradoxical nature of problems that seem simple (Goldbach’s Conjecture, for example.) and I love making things. (I’m a fan of oil painting and art in general.) Therefore, the reason I’m applying to graduate school is quite simple: to understand simple problems and create beautiful things.
In my algebra class on quivers this semester, my professor said something to the effect of: mathematics is about creating big theoretical frameworks in order to rephrase hard problems into easy ones. I’ve had some experience with this idea of a framework in the REU I participated in this past summer. We worked on classifying all graphs that had a 2D 2-contact representation. Succeeding in this effort, but discovering that it had been proved three years prior, we went on to abstract our paper to 3D 2-contact representations. However, our proof fell apart, and so we started from scratch to create a framework that proved the 3D case. Given the finite duration of the REU we never finished this task. Regardless of this, however, the experience gave me a feel for what it is to do research, and besides just going to graduate school to understand and create, I want to get my doctorate with the goal of becoming a mathematics professor to do research professionally.
Specifically, my interests lie in number theory and algebra. The simplicity of number theory problems fascinate me. In the independent study I took last semester in algebraic number theory, we discussed number fields, and how we can generalize the idea of a prime number to a prime ideal, and consequently study how prime ideals ramify or split in extensions of number fields, among other topics. It’s like when you just examine the integers you are looking at the outside of an intricate pocket-watch, but if you consider the algebraic aspect of it, you can open up the back cover and glimpse inside.
Additionally, I’m infatuated with how number theory pillages other areas of mathematics. In algebra, for instance, if you look at a proof of Sylow’s Theorem, you’ll find a lot of talk of fiddling with conjugacy classes and the like; and if you look at an analysis book, you’ll find a seemingly endless supply of epsilons and deltas. But if you open a number theory book (particularly algebraic), you can jump from using Minkowski 's Theorem to prove the finiteness of the Class Group via geometry to using Galois Theory to produce results about the splitting behavior of number fields, to playing with valuations and endowing topologies on algebraic structures! It’s very exciting!
These are the sort of things I want to continue studying. They’re so pure and conceptually simple at the surface, but when you dig down into them, you need to use every piece of mathematics you’ve ever learned in order to make sense of where you are. Throughout my undergraduate studies, I’ve been ambitious to learn and push myself whenever possible, and it’s all been so that I can study pure mathematics in graduate school and one day prove simple things like Goldbach’s Conjecture. It’s been considerably fun, and I hope I can continue my studies at the Universisty of <school name>.
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