Conference: CBMS Conference: Algorithmic Fractal Dimensions

该奖项资助了将于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数学科学(CBMS)区域研究会议“算法分形维”会议。会议的中心焦点将是爱荷华州立大学的杰克·卢茨教授将做的十场演讲。算法分维最早是在本世纪初发展起来的，用来衡量各种数学对象中的信息密度。顾名思义，它们是经典分维的版本，在它们的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了应用。最引人注目的是，算法分维正越来越频繁地被用来证明经典分形学中的新定理--其中一些回答了长期悬而未决的问题--这些定理的陈述不涉及可计算性或数理逻辑的相关方面。因此，来自多个研究界的与会者将对这次会议感兴趣。大会的邀请函将反映这种广度，并将优先考虑职业生涯早期的研究人员、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。会议讲座将从介绍经典的豪斯多夫和包装尺寸开始，并概述即将到来的讲座。接下来将讨论使数学概念生效意味着什么。然后，课程将继续使用有效性来定义经典Hausdorff和填充维度的算法版本。将讨论这些维度的应用，以及它们如何将Hausdorff维度与后来的Shannon和Kolmogorov的信息理论相统一。将强调使用算法维度来开发欧几里德空间中各个点的维度的有用概念，因为它导致了点到集原理，该原理使得几何测度论中有几个令人惊讶的应用。这些讲座还将讨论在计算复杂性、信息论和Borel正态数理论方面的应用。除了讲座之外，会议还将提供充足的时间进行公开问题会议和其他讨论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.