第二篇，數(shù)學(xué)專業(yè)。該生成功申請到了美國大學(xué)的Ph.D.in Mathematics全額獎(jiǎng)學(xué)金，選自《美嘉教育內(nèi)部PS范例》，請欣賞：

                                              Personal Statement

“The sole end of science is the honor of the human mind”——Carl Jacobi.

The first time I heard this words was when Professor Jianya Liu, dean of the Department of Mathematics of Shandong University, gave a speech on the history of prime numbers. A concise and vivid sketch of number theory was presented. When the famous talk of Jacobi was cited at the end of the speech, I was tremendously enlightened--I began to understand from the root level why we study mathematics.

I love math because it is elegant and natural. While I was attracted by the intriguing methods exerted in the text books, the ideas beneath the techniques engaged me the most. They compose the structure of theories, and theories are logically connected. In my opinion mathematics is a whole set of ideas instead of a collection of theorems and proofs. Most good ideas come naturally and simply.

Theorems, formulas and propositions are for future studies. I am good at holding them. The first step when I learn a theorem is to comprehend the ideas and ask what it originally dealt with. Sometimes this could be done by asking how the theorem makes sense in the simplest circumstance. The second step is to look into the proof, find whether there are shining points in the proof and how it comes. Some ingenious techniques make an appeal to me. Provided what a theorem exactly means is known, using the theorem becomes free and easy.

My ability of understanding is strengthened by grasping the backgrounds and ideas of theorems. This reading habit sharpened my sensitivity of interesting consequences. Comprehension of the ideas of predecessors develops my logic thinking and conjecturing skills. I often have good ideas in mind, and derive results from rigorous proof. This quality will be my advantage in the future research.

For example, when I was learning differential geometry, I always related the theory of higher dimensional to that of Euclidean space. And in algebra, mathematicians study Noetherian rings and Dedekind rings in order to derive properties of algebraic integers and these efforts facilitates the study of rational integers. Besides, I found most theories in functional analysis generalize the theories in mathematical analysis. Some theories are combinations and generalizations of mathematical analysis and linear algebra.

Research interest--number theory

I applied to the National Undergraduate Innovative Test Program funded by the Ministry of Education of the People's Republic of China. My advisor is Professor Zhiyong Zheng, who is the dean of our department, an expert in number theory. My purpose is to discuss the solvability of certain Diophantine equations. The project is still in progress. Up to now I have finished an essay on the topic of estimating the number of solutions of congruence with several variables.

Why am I interested in number theory? During the summer vacation 2009, I read Elementary Number Theory. Surprisingly, I found the theory of numbers is a beautiful art. With great passion, I did almost all the exercises listed in the book. The fascination of elementary theory motivated me to pursue modern theories.

To learn modern number theory, a solid and broad background in branches of mathematics is required. Since modern methods are utilized to solve classic problems, knowledge in algebra, complex analysis, geometry and topology is basic requirement for modern number theorists with diverse topics in mathematics interconnected. For instance, when Riemann discovered the relation between the distribution of the zeros of the Riemann zeta function and the distribution of the prime numbers, a delicate relation between discreteness and continuity has been established. Actually, this hypothesis serves as a bridge linking elementary number theory and analytic number theory.

Consequently, I am focusing on building a substantial foundation in algebra and reading a few topics in modern number theory. During the last year, I have been reading a couple of graduate text books published by Springer-Verlag, such as Lectures in Abstract Algebra written by Nanthan Jacobson, A Classical Introduction to Modern Number Theory written by Kenneth Ireland and Michael Rosen, and Algebra written by Serge Lang.

I always think about questions in number theory, and take energy solving problems. For instance, I have derived an explicit formula for a constant M, which was unsaid in a text book, to estimate the number of solutions of an equation over a finite field. A few days later, I amazingly found the same result with that in Andrew Weil’s paper. And when I was learning characters over finite fields, I invited a generalized definition of characters in a certain group.

Future plan

Having keen interest in number theory, I decide to be a number theorist and become a professor teaching at a university. I often consult my advisor Professor Zhiyong Zheng. In order to avoid misconception and to give a proper guidance, Professor Zheng has shown me what today’s number theorists are doing, and described the future of number theory. I learn that algebraic number theory and algebraic geometry are among the most popular sub-disciplines in number theory. Moreover, my reading schedule refers to Zheng’s suggestion.

The following is a carefully-arranged reading schedule.

N. Jacobson. Lectures in Abstract