Anabelian Geometry and Elementary Equivalence of Fields

阿纳贝尔几何和域的初等等价

• 批准号: 0401056
• 负责人: Florian Pop
• 金额: $ 15万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401056&HistoricalAwards=false
• 关键词: Anabelian; Geometry; Elementary; Equivalence; Fields

ABSTRACT for Award DMS-0401056 of PopThe present research project concerns modern Galois Theory,more specifically birational Anabelian Geometry. The mainproblem I will be investigating is trying to recover thebirational class of a variety (over an algebraically closedbase field) from the pro-l Galois theory of its function field.Related to this, there are several other questions, like theone by Ihara/Oda-Matsumoto concerning a geometric/combinatoricdescription of the Galois group of the field of rational numbers.The above question is also related to describing rational pointsof varieties via the so called Section Conjecture. This kind ofquestions were initiated -in the arithmetic situation- in someremarkable (unpublished) manuscripts by Grothendieck. But itappears that in higher dimensions, one has such "anabelianphenomena" even in the total absence of arithmetic, i.e., overan algebraically closed base field.The Galois Theory makes a bridge between two very differentmathematical aspects, namely some basic algebraic objects, likefields, or more general spaces (varieties) on the one side,and the way one solves algebraic equations, or constructscovers of the spaces in discussion, on the other side. Now the(birational) Anabelian Geometry asserts that in the case the fieldone works over is "primitive enough", respectively the geometryof the space one constructs covers of, is "complicated enough", thetotality of the "recipes" of solving all the algebraic equations,respectively of constructing all the covers, encodes the field,respectively the space under discussion. This opens a completelynew perspective in approaching some very fundamental mathematicalquestions. The present research project addresses some of the basicproblems in this mathematical field of research.

目前的研究项目涉及现代伽罗瓦理论，更具体地说，是双国安娜贝尔几何。我将研究的主要问题是试图从其函数场的亲伽罗瓦理论中恢复一个品种（在代数封闭基场上）的国家类。与此相关，还有其他几个问题，比如Ihara/Oda-Matsumoto关于有理数领域伽罗瓦群的几何/组合描述的问题。上述问题也与通过所谓的截面猜想来描述变种的有理点有关。这类问题是由格罗滕迪克（Grothendieck）在一些引人注目的（未发表的）手稿中提出的。但是，在更高的维度中，即使完全没有算术，也会出现这样的“易变现象”，即在代数封闭的基域上。伽罗瓦理论在两个非常不同的数学方面之间架起了一座桥梁，即一方面是一些基本的代数对象，如场，或更一般的空间（变种），另一方面是解决代数方程或构建讨论中的空间覆盖的方式。现在，（双分）安娜贝尔几何断言，如果所处理的域是“足够原始的”，那么所构造覆盖的空间的几何是“足够复杂的”，那么解决所有代数方程的“方法”的总和，分别是构造所有覆盖的“方法”，分别编码了所讨论的域和空间。这为解决一些非常基本的数学问题开辟了一个全新的视角。目前的研究项目涉及这一数学研究领域的一些基本问题。