Content and Readings

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.

Here are several suggested readings to consult in preparation for the meeting:

Who Asked Us? How the Theory of Computing Answers Questions about Analysis, by Jack Lutz and Neil Lutz

Algorithmic Fractal Dimensions in Geometric Measure Theory, by Jack Lutz and Elvira Mayordomo

Algorithmic Randomness (survey in Communications of the ACM), by Rod Downey and Denis Hirschfeldt

Computability and Randomness (survey in Notices of the AMS), by Rod Downey and Denis Hirschfeldt