**What is a statement of purpose?**

A statement of purpose (SOP or also called a statement of intent), in the context of applying for graduate schools or universities, is an essay that’s one of the most important aspects of your application because it tells the admission committee who you are, why you’re applying, why you’re a good candidate, and what you want to do in the future, your professional goals, what will you do when becoming alumni or PhD, apart from your GPA, test scores and other numbers. Therefore, don’t neglect the importance of this essay. It’s sometimes called an SOP letter, application essay, personal background, objectives for graduate study, cover letter, or something similar to one of these.

#### My Statement Of Purpose Majoring in Mathematics

**Learning Experiences**

I am a senior majoring in mathematics at Zhejiang University(ZJU). In 2009, I was admitted as a freshman to the Chu Kochen Honors College of ZJU, a place for elites designed to educate the top 5% of the students. Students in the CKC College faced intensive coursework and a very competitive environment since membership in the college is performance-dependent. Nevertheless, I did very well in all my classes, ranking 3rd among 269 in my first two years, and winning First Class Scholarship for two consecutive years. After careful consideration, I decided to follow my interest and chose mathematics as my major at the end of my first year.

**Student Seminars in ZJU: Leading Me on the Right Path**

During the summer of my freshman year, I signed up for two student seminars, Point Set Topology and Algebra organized by Prof. S.C. Wong of Zhejiang University. This was a turning point in my mathematical education, as I was gradually lead to the world of modern mathematics by them. In a typical seminar, regular lectures were given by students themselves instead of teachers. I signed for more than 5 talks per seminar, and my presentation skills gradually perfected. Moreover, from learnings in the seminars, I realized the importance of working out examples and problems. Not only do they solidify one’s current learnings, good examples often motivate further studies.

For instance, in a seminar using M.Artin’s *Algebra* as textbook, I came upon an example showing that the Riemann Surface for the algebraic equation y^2=x(x-1)(x-2) was a torus by explicitly computing its locus in P^2. However, when I tried to do the same for y^2=x(x-1)(x-2)(x-3), I found the locus becomes a torus with two points identified, and the original method fails. Naturally, I was very curious on knowing what the surface was for the latter equation. This motivated me in my later study of complex curves, where I found out that the correct answer is a torus using Riemann-Hurwitz formula. In fact, I recently learned a remedy for Artin’s method: By applying the “blowing up” surgery to the singular points of the locus(twice), we separates the ramified points and obtain the torus without singularity. The seminars continued throughout my second year, covering a wide range of topics listed in my CV. Together with the progression of problem solving and asking new questions, they have lead me on the right path of learning mathematics.

**Experience in Hong Kong:** **Efforts in Geometry and Topology**

During my third year from 2011.9 to 2012.6, I went to the University of Hong Kong as an exchange student with tuition paid by the Li&Fung Scholarship. This year brought up my interest and efforts in various areas of geometry, where I took advanced courses and studied more topics on my own.

First of all, motivated by learnings from the previous year, I attended a course in my first semester about Riemann Surfaces taught by Prof. Naming Mok of HKU. The lecture was for first-year graduate students, and I was the only third-year undergrad in class. During the semester, we went through the Riemann-Roch theorem and the Mittag-Lefler problem in the language of sheaf cohomology. The course gave me a good understanding on basic tools like divisors and sheaves, which provides me with a solid foundation for further study of complex algebraic geometry.

Another topic I studied intensively was algebraic topology . Throughout the year, I gradually built my foundations by working on the first three chapters of Hatcher’s *Algebraic Topology*. I also enrolled in two advanced undergrad courses in HKU, *Manifold Theory* by Prof. Si Ye Wu, and *Geometric Topology* by Prof P.P.Wong. Among other things, the courses each introduced its own homology theory, de Rahm and singular,completed the picture of homology theory from a topological viewpoint. I then intended to pursue a more rigorous study of homology from an algebraic viewpoint, so I also enrolled in a class on Homological Algebra. It turns out that the algebraic machinery can provide new perspectives to old problems. For example, after studying derived functors, I learned a sheaf theoretic proof of de Rahm’s theorem by constructing acyclic resolutions. This approach is conceptually cleaner, and can be copied to give new results such as the Dobeault Theorem.

I also learned Riemannian Geometry on my own in Hong Kong. Other than the basics, one of the most important techniques I learned was the use of moving frames. This method used by Chern is a powerful tool to analyzes the local data of a manifold, as I have seen from reading some research-level articles. For example, in an ongoing class taught by Prof. Hong Wei Xu , I studied papers by Yau on *pinching and classification of embeddings with constant mean curvature*, which used moving frames to obtain powerful results. I am certain that this will be of great assistance to my future learning.

My hard put efforts in geometry paid off. In the summer of 2012, I was selected as one of the 15 finalists to take the oral exam for the S.T.Yau’s College Math competition on the Geometry and Topology section. This further strengthened my confidence to take a further step in studying geometry.

**Future Planning**

I have since long made up my mind to go further on the training of becoming a mathematician. I have prepared myself well for my future graduate studies. Not only did I do well in my undergraduate courses, I also have plenty of undergraduate research experience listed in detail in my CV. I am clear on my interests and certain about my choice to continue my learnings.

My interests lie in many possible directions. Currently, differential geometry is my primary area of interest. I am especially interested in topics where geometry works together with other branches of mathematics. For one instance, I enjoyed my current learnings on Mean Curvature Flows, where techniques from PDE and geometry combine together to classify singularities and asymtotic behavior of flows. For another, I’m also keen on the topic of pinching problems and rigidity problems of submanifolds, which I’m currently taking a course on. Besides differential geometry, I also have a good interest in complex geometry. My prior readings on Forster, as well as well as the first two chapters of Griffiths and Harris’s book will give me a quicker grasp on the subject.

Ohio State University has one of the finest research groups in geometry in the world. It has many first-class mathematicians working in differential geometry and complex geometry like prof Fangyang Zhen. I hope that I would have the opportunity to study and make progress in Ohio State University.