Diophantine problems: new and old perspectives

数论是数学最古老的分支之一，近年来经历了一次复兴。一方面，安德鲁·怀尔斯对费马大定理的卓越证明，再次肯定了数论在现代数学中的中心地位。另一方面，在数据加密和信号传输等不同问题上的大量引人注目的应用已经证明了该领域在工业和政府目的上的实用性。数论中的一个基本问题是确定正属曲线上的积分点或有理点的数目和分布。在手头的项目中，研究者研究了丢番图方程子领域的一些相关问题，包括一些源于怀尔斯工作的问题。所采用的技术可以完全解决这一领域的许多经典问题。这些方程与纠错码理论有关，而纠错码反过来对诸如卫星、cd和dvd播放机、手机和智能卡的数据处理至关重要。

Number theory, one of the oldest branches of mathematics, has enjoyed a renaissance in recent years. On one hand, the remarkable proof of Fermat's Last Theorem by Andrew Wiles has reaffirmed Number theory's position as a central one in modern mathematics. On the other, numerous and striking applications to such diverse problems as data encryption and signal transmission have demonstrated the utility of the field, both for industrial and governmental purposes.One of the fundamental problems in Number  theory is that of determining the number and distribution of integral or rational points upon curves of positive genus. In the project at hand, the investigator studies a number of related questions in the subfield of Diophantine equations, including some stemming from work of Wiles. The techniques employed allow the complete resolution of a number of classic problems in this area. These equations are connected to the theory of error-correcting codes which, in turn, are of crucial importance to, for example, data processing from satellites, cd and dvd players, cell phones and smart cards.