Infinite Galois Theory in the Context of Hilbert's Tenth Problem

希尔伯特第十问题背景下的无限伽罗瓦理论

• 批准号: 2426399
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2426399
• 关键词: Infinite; Galois; Theory; Context; Hilbert

At the International Congress of Mathematicians in 1900, David Hilbert presented a list of 23 problems he deemed the most important questions in mathematics for the next century. Hilbert's Tenth Problem asks for an algorithm that decides for any given Diophantine equation if it has a solution in the integers or not. In particular, this algorithm would also be able to decide for any such equation if it has a solution in the rationals. Hilbert's Tenth Problem remained open until 1970 when Yuri Matiyasevich finally proved that such an algorithm could not exist. The methods introduced in his paper, however, only apply in the case of integer solutions, leaving the problem over the rationals open. In his original paper, Matiyasevich claims that Hilbert, who believed in the existence of an algorithm, would thus prob-ably not be satisfied with the current solution that only applies to the integers. He also points out that, at the time of writing in 1970, "progress in [the rational] case has been rather meagre." In recent years, progress has been made towards solving Hilbert's Tenth Problem over the rationals, in particular by Professor Koenigsmann's research group. In my research, I want to carry on the progress made so far and employ new techniques towards attacking the problem. In particu-lar, I want to draw from recent results in the areas of valuation theory, model theory, and infinite Galois theory, which are new and promising areas when it comes to applying them to Hilbert's Tenth Problem. Hilbert's Tenth Problem can naturally be translated to the language of model theory, amounting to the question of whether the existential first-order theory of the rational numbers is decidable or not. Moreover, methods from model theory have invoked more general valuation the-ory. This has led to a number of interesting new results and conjectures, among them the Elemen-tary Galois Conjecture (EGC). This conjecture suggests a deep connection between the absolute Galois group and the existence of certain henselian valuations, which would have far-reaching consequences. One such consequence provides evidence that the absolute Galois group over the rational numbers encodes sufficient information to answer Hilbert's Tenth Problem. At the mo-ment, however, this group is far from being fully understood, so my research will comprise a fur-ther investigation of this object in the context of Hilbert's Tenth Problem. The Elementary Galois Conjecture is also closely related to a famous, though not yet well understood conjecture, Grothendieck's Section Conjecture in his program called "anabelian geometry" from 1983. This project falls within the EPSRC Logic and Combinatorics research area.

在1900年的国际数学家大会上，大卫·希尔伯特提出了一个他认为是下一个世纪最重要的数学问题的23个问题的清单。希尔伯特第十问题要求一个算法，决定任何给定的丢番图方程，如果它有一个解决方案的整数或没有。特别是，这个算法也将能够决定任何这样的方程，如果它有一个解决方案的有理数。 希尔伯特第十问题一直悬而未决，直到1970年尤里·马蒂亚舍维奇终于证明了这样的算法不可能存在。在他的论文中介绍的方法，但是，只适用于整数解的情况下，离开了问题的合理开放。在他的原始论文中，Matiyasevich声称希尔伯特相信算法的存在，因此可能不会满足于仅适用于整数的当前解决方案。他还指出，在1970年撰写本书时，“[理性]案例的进展相当微弱。近年来，在解决希尔伯特第十问题的过程中取得了进展，特别是Koenigsmann教授的研究小组。在我的研究中，我想继续迄今为止取得的进展，并采用新的技术来解决这个问题。特别是，我想借鉴最近的结果，在领域的估值理论，模型理论和无限伽罗瓦理论，这是新的和有前途的领域，当谈到将它们应用到希尔伯特的第十问题。 希尔伯特的第十问题可以很自然地翻译成模型论的语言，相当于有理数的存在一阶理论是否可判定的问题。此外，模型论的方法已经调用了更一般的估值理论。这导致了许多有趣的新结果和新猜想，其中包括基本伽罗瓦猜想（EGC）。这个猜想暗示了绝对伽罗瓦群和某些亨塞尔赋值的存在之间的深刻联系，这将产生深远的影响。一个这样的结果提供了证据，证明有理数上的绝对伽罗瓦群编码了足够的信息来回答希尔伯特第十问题。然而，目前，这一群还远未被完全理解，因此我的研究将包括在希尔伯特第十问题的背景下对这一对象的进一步研究。基本伽罗瓦猜想也与一个著名的猜想密切相关，虽然还没有很好地理解，格罗滕迪克的部分猜想在他的程序称为“anabelian几何”从1983年。该项目属于EPSRC逻辑和组合学研究领域的福尔斯。