Geometric constructions over finite fields, elementary equivalence of finitely generated fields, and rational points on varieties

Bjorn M. Poonen研究者提出：1）用筛法证明有限域上各种几何对象和映射的存在性。 对于许多问题，标准的基于维数的论证只适用于无限域;我们的想法是用数值计数来代替它们。3）写一篇毕业论文，突出图式在非代数闭域上簇的研究中的应用。最初，代数几何关注的是多项式方程组的真实的数值解，但20世纪清楚地表明，通过考虑域来推广是富有成效的（即，数字系统）而不是真实的数字。 例如，有限域在纠错码和密码学理论中被证明是无价的。 不幸的是，许多几何构造只适用于无限域。 研究人员的第一个项目是使用组合技术，以适应这些结构的有限域的情况下。 研究者的第二个项目涉及不同领域的几何学之间的关系;它可能允许一个领域的结果转移到另一个领域。 最后，研究者建议写一个毕业论文，使学生能够使用这一领域新开发的技术。

DMS-0301280Bjorn M. PoonenThe investigator proposes:1) to use sieve methods to prove the existence of various geometricobjects and maps over finite fields. For many problems, the standarddimension-based arguments work only over infinite fields; the idea isto replace them by numerical counting.2) to continue the study initiated by F. Pop on the question ofwhether elementarily equivalent finitely generated fields arenecessarily isomorphic.3) to write a graduate text highlighting the application of schemes tothe study of varieties over nonalgebraically closed fields.Originally, algebraic geometry was concerned with the real numbersolutions to systems of polynomial equations, but the 20th centurymade it clear that it was fruitful to generalize by considering fields(i.e., number systems) other than the real numbers. For example,finite fields have proved invaluable in the theory of error-correctingcodes and cryptography. Unfortunately, many geometric constructionsare known to work only over infinite fields. The investigator's firstproject is to use combinatorial techniques to adapt theseconstructions to the finite field case. The investigator's secondproject concerns the relationships between geometry over differentfields; potentially it could allow results over one field to betransferred to another field. Finally, the investigator proposes towrite a graduate text that will make newly developed techniques inthis area accessible to students.