Computability and the absolute Galois group of the rational numbers

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。 然而，实际上，这一群体特别难以研究。 存在连续统--许多这样的对称性，其中大部分不能由任何计算机（或图灵机）运行任何有限长度的程序来计算。 然而，数学家们经常遇到的对称性基本上总是可计算的--也许是因为这些对称性是群的基础，或者仅仅是因为不可计算的对称性自然更难检验和处理。 这个项目旨在确定可计算对称（作为一个组）和所有对称的更大组之间有多大的差异。这项研究工作是在接口的逻辑和数论，很可能会吸引双方的利益。来自纽约市立大学研究生中心的研究生将参与这个项目。类似的情况也存在于所有真实的数的领域：只有可数的多个真实的数具有可计算的小数展开式，因此绝大多数真实的数是不可计算的，然而可计算的数是日常生活中唯一遇到的数。 这里，已知可计算的真实的数形成与所有真实的数的全域极其相似的子域，具有完全相同的一阶性质的基本子域。该基金将资助研究，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？ (Or至少，这两个基本上是等价的吗？） 如果是这样的话，那么数学家们应该能够仅仅通过考察可计算对称性来确定关于全群的许多结果，这要容易得多。 如果不是，那就意味着绝对伽罗瓦群是一个比真实的数域更棘手的对象，因为它的不可计算的对称性在某种程度上对它的性质至关重要。 然而，即使这样，对于相对简单的属性（例如，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.