Conference: CBMS Conference: Algorithmic Fractal Dimensions

该奖项资助了数学科学（CBMS）区域研究会议的会议委员会将于2024年5月20日至24日在爱荷华州得梅因的德雷克大学举行的“分形维数”。 会议的中心议题是由爱荷华州州立大学的杰克·卢茨教授作的十场演讲。 数学分形维数最早是在本世纪初作为各种数学对象中信息密度的度量而发展起来的。 顾名思义，它们是经典分形维数的版本，其定义中嵌入了计算理论。 它们已经在可计算性理论、计算复杂性、信息论和数论中得到应用。 最引人注目的是，算法分形维数越来越频繁地被用来证明新的定理-一些回答长期存在的开放问题-在经典分形几何中，定理的陈述不涉及可计算性或相关方面的数学逻辑。 因此，会议将吸引来自多个研究社区的与会者。 会议邀请函将反映这种广度，并将优先考虑早期职业研究人员，历史上代表性不足的群体的成员，以及中西部的一系列机构。会议讲座将开始，介绍经典豪斯多夫和包装尺寸和讲座的概述。 接下来将讨论有效化一个数学概念意味着什么。 然后，讲座将继续使用有效化来定义经典豪斯多夫和包装尺寸的算法版本。 这些尺寸的应用将被讨论，沿着与他们如何统一Hausdorff维数与后来的信息理论的香农和Kolmogorov。 将强调使用算法维数来发展欧几里得空间中各个点的维数的有用概念，因为它导致了点到集原理，该原理在几何测度理论中有几个令人惊讶的应用。 讲座还将讨论在计算复杂性，信息论和博雷尔正常数理论的应用。 除了讲座之外，会议还将提供充足的时间进行开放问题会议和其他讨论。该奖项反映了NSF的法定使命，并被认为值得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来支持。

This award funds a Conference Board of the Mathematical Sciences (CBMS) Regional Research Conference on "Algorithmic Fractal Dimensions" to be held May 20-24, 2024, at Drake University in Des Moines, Iowa. The central focus of the conference will be ten lectures to be given by Professor Jack Lutz of Iowa State University. Algorithmic fractal dimensions were first developed at the beginning of this century as measures of the density of information in various mathematical objects. As the name suggests, they are versions of classical fractal dimensions that have the theory of computing embedded in their definitions. They have already had applications in computability theory, computational complexity, information theory, and number theory. Most strikingly, algorithmic fractal dimensions are being used with increasing frequency to prove new theorems--some answering long-standing open problems--in classical fractal geometry, theorems whose statements do not involve computability or related aspects of mathematical logic. The conference will thus be of interest to participants from multiple research communities. Invitations to the conference will reflect this breadth and will prioritize early-career researchers, members of historically underrepresented groups, and a range of institutions in the Midwest.The conference lectures will begin with an introduction to the classical Hausdorff and packing dimensions and an overview of the lectures to come. Following that will be a discussion of what it means to effectivize a mathematical concept. The lectures will then proceed to use effectivization to define algorithmic versions of classical Hausdorff and packing dimensions. Application of these dimensions will be discussed, along with how they unify Hausdorff dimension with the later information theories of Shannon and Kolmogorov. The use of algorithmic dimensions to develop useful notions of the dimensions of individual points in Euclidean space will be emphasized, as it leads to the Point-to-Set Principle that has enabled several surprising applications in geometric measure theory. The lectures will also discuss applications in computational complexity, information theory, and the theory of Borel normal numbers. Beyond the lectures, the conference will provide ample time for open problem sessions and other discussions.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.