Additive and probabilistic number theory in function fields

初等数论研究的是整数的算术性质。由于有限域（称为函数域）上的多项式环与整数环非常相似，许多适用于整数的结果也适用于多项式。我的研究方向是函数场中的加性和概率数论。由于多项式中有比整数中更多的结构可供研究，函数领域的工作已经成为数论其他分支研究的刺激。另一方面，由于基域的特征是正的，许多用于整数的方法不能直接应用于多项式。由于经常需要新的方法，函数域中的数论因其本身的缘故而引起人们的兴趣。此外，函数场理论不过是代数曲线理论的另一种伪装。后者在费马大定理的证明中起着至关重要的作用。它也有许多实际应用，例如在现代密码学中使用椭圆曲线。我想进一步探讨和阐述函数领域数论与其他数学领域的联系。

Elementary number theory is concerned with the arithmetic properties of integers. As the ring of polynomials over a finite field, called a function field, bears a close resemblance to the ring of integers, many results which hold for integers have analogues for polynomials. My research is to study additive and  probabilistic number theory in function fields. Since there are more structures in polynomials than in integers which one can play with, work in function fields has served as a stimulus to research in other branches of number theory. On the other hand, since the characteristic of the base field is positive, many methods used for integers cannot be applied directly to polynomials. As new approaches are often required, number theory in function fields is of interest for its own sake. Moreover, the theory of function fields is but another guise for the theory of algebraic curves. The latter plays an essential role  in the proof of Fermat's Last Theorem. It also has many real-world applications, such as the use of elliptic curves in modern cryptography. I would like to further explore and elaborate upon the connections of  number theory in function fields to other areas of mathematics.