Information Coded in Mathematical Structures
以数学结构编码的信息
基本信息
- 批准号:2419591
- 负责人:
- 金额:$ 17.78万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2023
- 资助国家:美国
- 起止时间:2023-11-01 至 2025-08-31
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
Mathematical logic uses mathematical tools in an introspective way to study mathematics itself. This project will use the tools of mathematical logic (and computability theory in particular) to study the way that information can be coded into mathematical structures of the types that arise in all areas of mathematics. The general paradigm is that information is encoded into a mathematical structure if it can always be recovered in an intrinsic way from the structure, without artifacts from the way that the structure is presented. For the simplest kinds of information, like a string of 0's and 1's, the situation is well-understood. But for more complex kinds of information, the situation is much less well-understood and there are many interesting phenomena to explore. Understanding the coding of information will help us understand both the nature of information and the ability of structures of various kinds to code information. This, in turn, informs and guides the mathematical practice of other mathematicians. This project includes the training of undergraduate and graduate students and outreach to high schools.More formally, we say that a piece of information A is coded in a structure B if from every copy of the structure, we can recover in a computable way a copy of the information A. For example if B is a countably infinite group then a copy of B is a Cayley table for the group, noting that for an infinite group there are many different Cayley tables obtained by listing the elements of the group in different orders. By a piece of information we mean, in order of increasing complexity, a binary string (or subset of the natural numbers), an infinite family of binary strings, an infinite tree, or some other structure. For the first case of a binary string, there is a good structural characterisation of when a structure codes a binary string; one can interpret this as saying that there is always a good reason for a binary string to be coded by a structure. This is not the case for any more of the more complicated types of information; it seems that there are structures which happen to code information, but there is no good structural reason as to why. This project aims to explore these phenomena and push them to their limit with one aim being to give new tools to attack difficult open problems on degree spectra which have been resistant to current techniques.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.
数理逻辑以内省的方式使用数学工具来研究数学本身。该项目将使用数理逻辑(特别是可计算性理论)的工具来研究信息如何被编码到所有数学领域中出现的数学结构中。一般的范式是,信息被编码到一个数学结构中,如果它总是可以从结构中以内在的方式恢复,而没有结构呈现方式的伪像。对于最简单的信息类型,例如0和1的字符串,这种情况很容易理解。但是对于更复杂的信息,情况就不那么好理解了,有许多有趣的现象需要探索。理解信息的编码将帮助我们理解信息的本质和各种结构编码信息的能力。这反过来又通知和指导其他数学家的数学实践。这个项目包括对本科生和研究生的培训,以及对高中的推广。更正式地说,我们说一段信息A被编码在一个结构B中,如果从这个结构的每一个副本中,我们可以用可计算的方式恢复信息A的一个副本。例如,如果B是一个可数无限群,那么B的一个副本是这个群的一个凯莱表,注意对于一个无限群,有许多不同的凯莱表,通过以不同的顺序列出这个群的元素来获得。我们所说的一条信息,按照复杂性的增加顺序,是指一个二进制字符串(或自然数的子集)、一个无限的二进制字符串族、一棵无限树或其他一些结构。对于二进制字符串的第一种情况,当一个结构编码一个二进制字符串时,有一个很好的结构特征;人们可以将其解释为总是有一个很好的理由让一个二进制字符串被一个结构编码。对于任何更复杂的信息类型,情况就不是这样了;似乎有一些结构碰巧对信息进行编码,但没有很好的结构理由来解释为什么。该项目旨在探索这些现象,并将其推向极限,其中一个目的是提供新的工具来解决度谱上的困难开放问题,这些问题对当前技术具有抵抗力。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。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Matthew Harrison-Trainor其他文献
The logic of cardinality comparison without the axiom of choice
没有选择公理的基数比较的逻辑
- DOI:
10.1016/j.apal.2024.103549 - 发表时间:
2025-04-01 - 期刊:
- 影响因子:0.600
- 作者:
Matthew Harrison-Trainor;Dhruv Kulshreshtha - 通讯作者:
Dhruv Kulshreshtha
A note on cancellation axioms for comparative probability
- DOI:
10.1007/s11238-015-9491-2 - 发表时间:
2015-03-21 - 期刊:
- 影响因子:0.600
- 作者:
Matthew Harrison-Trainor;Wesley H. Holliday;Thomas F. Icard - 通讯作者:
Thomas F. Icard
Matthew Harrison-Trainor的其他文献
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{{ truncateString('Matthew Harrison-Trainor', 18)}}的其他基金
Information Coded in Mathematical Structures
以数学结构编码的信息
- 批准号:
2153823 - 财政年份:2022
- 资助金额:
$ 17.78万 - 项目类别:
Standard Grant
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