Computability Theory and Algebraic Structures

可计算性理论和代数结构

基本信息

  • 批准号:
    0704256
  • 负责人:
  • 金额:
    $ 5.14万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2007
  • 资助国家:
    美国
  • 起止时间:
    2007-07-01 至 2010-03-31
  • 项目状态:
    已结题

项目摘要

Harizanov integrates the ideas and methods from computability theory, model theory, algebra, and topology to investigate algorithmic phenomena on algebraic structures. She aims to better understand how algorithmic properties interact with algebraic and topological ones. Harizanov focuses on the complexity of countable models, their submodels and relations, and transformations of structures. She studies model-theoretic complexity of computable structures measured by their Scott rank. Harizanov investigates intrinsic complexity of relations on structures, captured both syntactically and by their Turing or strong degree spectra. She relates degree spectra of structures to the degree spectra of relations via spectrally universal structures, which are often obtained as Fraisse limits. Harizanov investigates the spaces of left orders and bi-orders on magmas, semigroups, and groups of importance in low-dimensional topology and algebra. She connects their topological properties with computability-theoretic ones. Harizanov also studies algorithmic complexity of isomorphisms. This includes effective categoricity of specific and general computable structures. It also includes the study of partial and total automorphisms. One direction is to develop the theory of automorphism degree spectra of computable structures. The other direction is to investigate various semigroups of partial automorphisms and how the structures can be recovered from these semigroups. Invented in the 1930's, computability theory paved the way for the creation of the modern programmable computer. The main thrust of the field is to understand the limitations on algorithmic computation, without regard for the physical implementation. Interaction of computability theory with model theory and universal algebra, as well as other areas of mathematics, has resulted in computable model theory and, more generally, in computable mathematics, a very active research area in the last few decades. Harizanov's proposed research includes a broad range of topics in computable mathematics. She investigates when some mathematical constructions are algorithmic, or can be replaced by algorithmic ones yielding the same results, and when they are fundamentally nonalgorithmic. Undecidable sets, as well as problems these sets encode, can be more precisely classified by considering algorithms with oracles, which require external knowledge. Turing and other computability-theoretic degrees, which play a significant role in Harizanov's project, provide an important measure of the level of such knowledge needed. Harizanov combines degree-theoretic and other unique methods from computability theory with algebraic, combinatorial and topological methods to study and explain phenomena of importance in other areas of mathematics.
Harizanov 综合了可计算性理论、模型论、代数和拓扑学的思想和方法来研究代数结构上的算法现象。她的目标是更好地理解算法属性如何与代数和拓扑属性相互作用。 Harizanov 专注于可数模型的复杂性、它们的子模型和关系以及结构的转换。她研究通过斯科特等级衡量的可计算结构的模型理论复杂性。哈里扎诺夫研究了结构关系的内在复杂性,通过句法和图灵或强度谱来捕获。她通过谱通用结构将结构的度谱与关系的度谱联系起来,这些结构通常作为弗赖斯极限获得。 Harizanov 研究了岩浆、半群以及低维拓扑和代数中重要群的左阶和二阶空间。她将它们的拓扑性质与可计算性理论联系起来。哈里扎诺夫还研究同构的算法复杂性。这包括特定和一般可计算结构的有效分类。它还包括部分和完全自同构的研究。一个方向是发展可计算结构的自同构度谱理论。另一个方向是研究部分自同构的各种半群以及如何从这些半群中恢复结构。可计算性理论发明于 1930 年代,为现代可编程计算机的创建铺平了道路。该领域的主要目标是了解算法计算的局限性,而不考虑物理实现。可计算性理论与模型论和泛代数以及其他数学领域的相互作用产生了可计算模型理论,更广泛地说,在可计算数学中,这是过去几十年来一个非常活跃的研究领域。哈里扎诺夫提出的研究涵盖了可计算数学领域的广泛主题。她研究了某些数学结构何时是算法性的,或者可以被产生相同结果的算法结构所取代,以及何时它们基本上是非算法的。不可判定的集合以及这些集合编码的问题可以通过考虑带有预言机的算法来更精确地分类,这需要外部知识。图灵和其他可计算性理论学位在哈里扎诺夫的项目中发挥着重要作用,提供了衡量所需知识水平的重要衡量标准。哈里扎诺夫将度理论和可计算性理论中的其他独特方法与代数、组合和拓扑方法相结合,以研究和解释数学其他领域的重要现象。

项目成果

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Valentina Harizanov其他文献

Valentina Harizanov的其他文献

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{{ truncateString('Valentina Harizanov', 18)}}的其他基金

FRG: Collaborative Research: Definability and Computability over Arithmetically Significant Fields
FRG:协作研究:算术上重要字段的可定义性和可计算性
  • 批准号:
    2152095
  • 财政年份:
    2022
  • 资助金额:
    $ 5.14万
  • 项目类别:
    Standard Grant
Topics in Computable Structure Theory
可计算结构理论专题
  • 批准号:
    1202328
  • 财政年份:
    2012
  • 资助金额:
    $ 5.14万
  • 项目类别:
    Standard Grant
Topics in Computable Mathematics
可计算数学主题
  • 批准号:
    0904101
  • 财政年份:
    2009
  • 资助金额:
    $ 5.14万
  • 项目类别:
    Standard Grant
Computability Theory and Algebraic Structures
可计算性理论和代数结构
  • 批准号:
    0502499
  • 财政年份:
    2005
  • 资助金额:
    $ 5.14万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Frequency Approach to Approximating Algorithms
数学科学:近似算法的频率方法
  • 批准号:
    9210443
  • 财政年份:
    1992
  • 资助金额:
    $ 5.14万
  • 项目类别:
    Standard Grant

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