My Statement Of Purpose as a Mathematics PhD Student

What Is a Statement of Purpose

A statement of purpose is a written statement composed in the standard essay format. It describes a student’s motivation for applying to graduate school, demonstrates their knowledge and experience, and gives the admissions office an accurate portrayal of who the student is as a person. It is always useful to include long-term goals and ways of achieving them, such as graduating from school. The office of admissions does not know you, and how well you explain who you are and why you are the right person to be admitted is up to you. Your statement of purpose is the best tool for realizing these intentions. 

My Statement Of Purpose as a Mathematics PhD Student

Since entering college as a mathematics major, I’ve suspected that I would pursue a PhD in mathematics. Over the past four years at Oberlin College, my experiences have helped solidify these plans. I’ve sought ways to extend my education outside of Oberlin’s normal course offerings, through independent study and the Budapest Semesters in Mathematics program. I’ve also had two summer opportunities to prepare myself for graduate school, at the Cornell University Summer Math Institute and the Williams College SMALL Research Experience for Undergraduates. These experiences have challenged and excited me, and prepared me to pursue a future in mathematics.

My first exposure to academia came the summer after my sophomore year, at the Cornell Summer Math Institute. Before this point, I knew that I enjoyed mathematics, but I had very little exposure to what might lie in my future. Within two days of arriving at Cornell, all of my previous notions of graduate school and academia were broken down, and I was left questioning myself and what I wanted. It took the next eight weeks to rebuild my self confidence. We spent each day attending an algebra course and working on research projects in functional analysis. The schedule was demanding, but not impossible; it was intended as a rough estimate of the life of a graduate student. As a student at a small liberal arts college, this experience was invaluable. At Cornell, I was able to talk with current graduate students and hear what they had to say about graduate school, both good and bad. Their advice, warnings, and anecdotes reshaped my understanding of what graduate study entails. As I developed a stronger idea of the demands of graduate school, I reevaluated myself and my goals. By the end of the summer, graduate study was no longer an uncertain possibility for my future, but a concrete plan.

A few short weeks after leaving Cornell I arrived in Budapest. Following my summer in Ithaca, I was eager to take a class load that wouldn’t be possible at Oberlin, both in number and rigor of courses. Many of the courses were difficult, yet engaging, and presented mathematics from a new perspective. It was wonderful to escape the boundaries of Oberlin’s small department and take classes from different professors. The semester challenged me, academically and personally, but I finished it confident in my abilities. My experience at Cornell showed me what graduate study demands; that fall taught me I am capable of it.

My first real research experience came this summer in the Williams College SMALL REU. My group’s project was on Algebraic and Geometric Combinatorics, advised by Elizabeth Beazley, a topic much closer in line with my mathematical interests than functional analysis. Our goal was to generalize a projection on the affine Grassmannian due to Berg, Jones, and Vazirani to one on parabolic quotients of affine Weyl groups in other classical Lie types. The original projection is described as a bijection in several different models for realizing the elements of the affine Grassmannian, both combinatorial and geometric, which have analogues in other the other types. We were able to successfully generalize the projection to type C, interpreting the map in each of the models. Many of the results of our research are available in the paper we wrote, Bijective Projections on Affine Weyl Groups, available on arXiv at ; section 6 in particular highlights my contributions. Throughout the summer I focused my work on the geometric model with great success. We were able to geometrically realize the bijection as an explicit projection between alcoves, which allowed us to prove deeper properties of the projection beyond those evident from the combinatorics alone. A number of our results for type C extend beyond the scope of the paper we were hoping to generalize.

Besides writing a paper, our group has focused on presenting our research at conferences. In late July, we gave a talk at the Young Mathematicians Conference at The Ohio State University and earned second prize for the quality and content of our presentation. At YMC, I was able to meet mathematics students from around the country, as well as get a feel for OSU’s math program. Besides attending at YMC, we are presenting again in January at the Joint Mathematics Meetings, and we’ve also applied to the Formal Power Series and Algebraic Combinatorics conference next June in Paris, France.

My summer at SMALL was more than just nine weeks of research. It was also nine weeks which I spent surrounded by top mathematics students from around the country. I was awed by their caliber, but also by my ability to hold my own working with them. While I had not necessarily taken the same level of coursework as some of my peers, what I knew, I knew well, and I was able to quickly learn anything I was missing. As a researcher, I was among equals. This reassured me that despite coming from a small college with no graduate program, I am well prepared to succeed in graduate school among students who benefited from opportunities which I lacked, such as the chance to take graduate courses. This experience also gave me the chance to engage in real, meaningful research. Unlike the previous summer, I felt there were useful applications to our work and that it was not far off from research I might do in graduate school. The most important lessons I learned this summer were not specific to the problem we were working on, but rather pertinent to mathematical research in any area. This summer taught me how to better effectively engage with a problem and make progress toward a solution. Most importantly, I learned that mathematical research is something I enjoy and want to keep doing.

I have also sought opportunities to extend my Oberlin education past the course offerings, which I’ve exhausted. I’ve engaged in independent study in three different contexts in the past year, studying combinatorics during Oberlin’s Winter Term and representation theory of the symmetric group as a private reading last spring, and this fall I was invited to do an honors project under the supervision of Susan Jane Colley. For my project, I am studying commutative algebra at the graduate level. I’ve found these opportunities for independent study beneficial in a number of ways: I’ve learned interesting and useful material, gained the ability to learn independently, and practiced communicating mathematics through presentations of what I’ve learned to my advising professor.

I am drawn to OSU not only because of the strength and size of the program as a whole, but also because of its strength in my field of choice, algebraic geometry, and related fields. I’m drawn to algebraic geometry due to the deep relationship between algebra and geometry, which compares with the interplay between combinatorics, algebra, and geometry in my summer research, and I’m eager to engage in research on my own. I think OSU would be an excellent place for me to do this, especially coming from a small liberal arts college; the size and breadth of the department would expose me to fields I didn’t encounter as an undergraduate. I’ve worked hard to prepare myself for graduate study and I’m excited to begin this next step.