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.

绝对伽罗瓦群 Gal(Q) 在整个数学中是众所周知的。 它的元素正是有理数的代数闭包的对称性。 但实际上，这个群体特别难以研究。 这些对称性有很多连续体，其中大多数都不能由任何运行任何有限长度程序的计算机（或图灵机）来计算。 然而，数学家经常遇到的对称性本质上总是可计算的——也许是因为它们是该群的基础，或者可能只是因为不可计算的对称性自然更难以检查和使用。 该项目旨在确定可计算的对称性（作为一个群）与所有对称性的较大群之间有多少差异。这项研究工作处于逻辑和数论的交叉领域，很可能会引起两个领域的兴趣。纽约市立大学研究生中心的研究生将参与该项目。所有实数领域也存在类似的情况：只有可数个实数具有可计算的十进制展开式，因此绝大多数实数是不可计算的，而可计算的实数是日常生活中唯一遇到的。 在这里，我们知道可计算实数形成了一个与所有实数的全域极其相似的子域，一个具有完全相同的一阶性质的基本子域。这笔赠款将资助研究，试图确定 Gal(Q) 是否以这种方式类似：可计算对称性是否形成全群的基本子群？ （或者，至少，这两者在本质上是等价的？）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整个群的许多结果，而这些对称性更容易获得。 如果不是，那就表明绝对伽罗瓦群是一个比实数域更棘手的对象，其不可计算的对称性在某种程度上对其特征至关重要。 然而，即使如此，对于相对简单的属性（例如，关于该群的纯粹存在性陈述），子群也可能是基本的，在这种情况下，该项目将尝试找到子群停止模仿整个群的第一个级别该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。