Each applicant must submit an academic statement of purpose (ASOP). The ASOP is one of your primary opportunities to help the admissions committee understand your academic objectives and determine if you are a good match for the program you are applying to. The goal of this document is to impress upon the admissions committee that you have a solid background and experience in your area of interest and that you have the potential to be successful in graduate study.
The ASOP is also a place, if necessary, where you can (and should) address any blemishes, gaps, or weaknesses in your academic record. In these situations, you will want to be honest, but brief. It is best to turn negatives into positives by focusing on how you overcame obstacles, remained persistent in the pursuit of your goals, and showed resilience. Share what you learned from the particular experience, and how it led you to become a better researcher/scholar/person, etc.
Statement Of Purpose Sample
I am an aspiring number theorist. Motivated by a desire to see the big picture of mathematics, I have pursued mathematics seriously since the fall of 2007. As a leader in research and education, Ohio State University oﬀers the ideal environment for me to continue my studies as a graduate student. Abundant coursework, research experience, and years of tutoring give me confidence that I will thrive in Ohio State’s rigorous academics.
At Amherst College, I took 19 courses in mathematics, participated in 4 research projects, and tutored for 3 years in most math subjects. Professor Benjamin Hutz advised me on the first research project, in arithmetic dynamics, during the summer after my first year of college. I computed a uniform bound on the rational, iterated pre-images of -1 under the family of quadratic polynomials over Q using heights on elliptic curves. My result was published in . The success of this project led us to a generalized uniform bound for the rational, iterated pre-images of Q under quadratic polynomials over Q. We published the results in .
In the summer of 2011, I participated in an NSF funded REU at Amherst College under the guidance of Professor Robert Benedetto in the field of arithmetic dynamics. We collected dynamical data  for quadratic polynomials over Q. I resumed work with Prof. Benedetto in the Fall of 2012 to implement our computational technique for quadratic rational functions defined over Q—the work is ongoing.
Professor David A. Cox advised me on a senior honours thesis in which I investigated the Galois theory of the lemniscate.1 One may construct so-called lemnatomic extensions of Q(I) by adjoining division points of the lemniscate much in the same way one constructs cyclotomic extensions of Q by adjoining division points of the circle. The Galois groups of such extensions had been studied by Abel and others in their relation to ruler and compass constructions, but the group had not been fully determined. By developing a theory of lemnatomic polynomials akin to their cyclotomic cousins, I determined, with proof, the Galois groups of lemnatomic extensions.
I presented my findings at the SUMS, GSUMC, and HRUMC undergraduate math conferences in the spring of 2012. For my work, Amherst College awarded me the Robert Breusch Prize2, a Post-Baccalaureate Fellowship to continue research the following summer, and Summa Cum Laude. Together, Professor Cox and I found another proof of my main result as well as other original discoveries. We collected our work in a paper and submitted it for publication; a preprint may be found at .
My interest in number theory began in 2007 while reading a book entitled Excursions in Number Theory. Upon matriculation at Amherst College, I registered for the introductory number theory course. Quadratic reciprocity, Gaussian integers, and the classification of primes expressible as the sum of squares inspired me to learn more. Prof. Hutz introduced me to elliptic curves and heights during my summer project in 2009. Subsequent courses in algebra, algebraic geometry and complex analysis prepared me for a more advanced study of arithmetic. When Professor Djordje Mili´cevi´ joined the Amherst faculty for the 2010-11 academic year, I studied analytic number theory with him in a special topics course. David Cox taught me Galois theory, which led to the senior honours project discussed above. In senior year I devoted most of my time to the study of algebraic number theory from Marcus’s Number Fields and Cox’s Primes of the Form X2 + NY2. I concluded my time at Amherst with a course on group representation theory.
Following graduation, I continued my studies in number theory and geometry. My initial curiosity about the Gaussian integers developed into a fascination with the structure of number fields, and in particular, the insights obtained via class field theory. Thus I have been working on Iwasawa’s Local Class Field Theory and Silverman’s Arithmetic of Elliptic Curves. Other projects include digesting all the wonderful expository pieces available on Keith Conrad’s web page and pursuing further studies of the lemnatomic extensions. One nice result of this research is a generalized Wallis product related to the family of clover curves—I plan to submit the paper for publication once finished with graduate applications.
If admitted to Ohio State University, I would like to study algebraic number theory and the arithmetic of function fields with Professor David Goss. After consulting my advisers and reviewing his publications, I feel that our interests are well-aligned. In addition, the large and diverse department will oﬀer the opportunity to explore a variety of fields and expand my breadth. The active research seminar in number theory indicates a large and vibrant mathematical community in which I would very much like to participate. On a personal note, after my time on the East coast as an undergraduate, I would be delighted to return to the midwest in closer proximity to my family.
I began work as full-time math, physics, and computer science tutor in the fall of 2012 at Amherst College’s Moss Quantitative Center. My studies and research are ongoing; I continue to work closely with the faculty at Amherst as well as regularly attend the weekly Pioneer Valley Number Theory Seminar to hear about recent developments in the field. I decided to not proceed directly to graduate school from Amherst for two reasons: one, between coursework and a thesis, I did not feel that I had enough time to give the application process the thought and eﬀort it required; two, the Moss Quantitative Fellowship oﬀered a great opportunity to simultaneously develop my teaching skills, master the undergraduate curriculum, and provide the time and resources for self-study. Having just completed the first semester of my life since age 5, not in school, I feel rejuvenated and excited to begin a doctoral program. Come fall, I will have an extra year of mathematics and maturity under my belt for the challenges to come.
Thus, considering my background, experience, and motivation, I feel prepared and qualified to join the Ohio State University community.