Computability and the absolute Galois group of the rational numbers

可计算性和有理数的绝对伽罗瓦群

• 批准号: 2348891
• 负责人: Russell Miller
• 金额: $ 19.5万
• 依托单位: CUNY Queens College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348891&HistoricalAwards=false
• 关键词: Computability; absolute; Galois; group; rational

The absolute Galois group Gal(Q) is well-known throughout mathematics. Its elements are precisely the symmetries of the algebraic closure of the rational numbers. In practice, though, this group is particularly difficult to study. There are continuum-many of these symmetries, most of which cannot be computed by any computer (or Turing machine) running any finite-length program whatsoever. However, the symmetries that mathematicians encounter on a regular basis are essentially always computable -- perhaps because these are fundamental to the group, or perhaps just because noncomputable symmetries are naturally more difficult to examine and work with. This project aims to determine just how much difference there is between the computable symmetries (as a group) and the larger group of all symmetries. The research work lies at the interface of logic and number theory and is likely to attract the interest of both communities. Graduate students from CUNY Graduate Center will participate in this project. An analogous situation exists with the field of all real numbers: only countably many real numbers have computable decimal expansions, so the vast majority of real numbers are noncomputable, yet the computable ones are the only ones ever encountered in daily life. Here, it is known that the computable real numbers form a subfield extremely similar to the full field of all real numbers, an elementary subfield with exactly the same first-order properties. This grant will fund research to attempt to determine whether Gal(Q) is analogous in this way: do the computable symmetries form an elementary subgroup of the full group? (Or, at a minimum, are the two elementarily equivalent?) If so, then mathematicians should be able to determine many results about the full group just by examining the computable symmetries, which are far more accessible. If not, that would suggest that the absolute Galois group is a thornier object than the field of real numbers, with its noncomputable symmetries somehow essential to its character. However, even then, it is possible that the subgroup might be elementary for relatively simple properties (e.g., purely existential statements about the group), in which case this project will attempt to find the first level at which the subgroup stops imitating the full group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

绝对的Galois组GAL（Q）在整个数学中都是众所周知的。 它的元素恰恰是理性数字的代数关闭的对称性。 但是，实际上，这个小组特别难以研究。 这些对称性有许多连续的对称性，其中大多数无法由运行任何有限长度程序的任何计算机（或图灵机）计算。 但是，数学家定期遇到的对称性基本上是可以计算的 - 也许是因为这些对群体至关重要，或者仅仅是因为不合格的对称性自然更难以检查和工作。 该项目旨在确定可计算对称性（作为一个组）和所有对称组的较大组之间有多大差异。研究工作在于逻辑和数理论的界面，很可能会吸引两个社区的兴趣。来自CUNY研究生中心的研究生将参加该项目。与所有实际数字的领域相似：只有许多实数具有可计算的小数扩展，因此绝大多数实数都是不可归计的，但是可计算的数字是日常生活中唯一遇到的次数。 在这里，众所周知，可计算的实数形成一个与所有实数的完整字段极为相似的子字段，该字段是具有完全相同的一阶属性的基本子场。该赠款将资助研究以尝试确定GAL（Q）是否以这种方式相似：可计算对称性是否形成了整个组的基本亚组？ （或者，至少是两个基本等效的吗？）如果是这样，那么数学家应该能够通过检查可计算的对称性来确定有关完整组的许多结果，这些对称性更容易访问。 如果不是这样，那将表明绝对的Galois群是一个比实数领域的棘手对象，其不可归结的对称性对其性格必不可少。 但是，即使到那时，子组也可能是相对简单的属性（例如，纯粹的存在的陈述）是基本的，在这种情况下，该项目将试图找到第一个子组停止模仿完整组的第一个级别。该奖项反映了NSF的法定任务，并通过评估了基金会的智力效果，并通过评估了基金会的范围。