Instantaneous Dimension of Metric Spaces via Spread and Magnitude

Explore the concept of instantaneous dimension in metric spaces through a 51-minute lecture by Simon Willerton from the Applied Algebraic Topology Network. Delve into the intriguing idea that spaces can exhibit different dimensions at various scales, using examples like a long thin strip to illustrate this phenomenon. Learn about quantifying these dimensional changes through the growth rate of size in metric spaces. Discover the notion of 'magnitude' introduced by Leinster using category theoretic ideas, and its connections to diverse areas such as biodiversity and potential theory. Examine the related concept of 'spreads' and its computational simplicity. Investigate Meckes' findings on the relationship between asymptotic growth rate of magnitude and Minkowski dimension. Gain insights into non-asymptotic growth rates and their interesting implications for finite metric spaces, revealing scale-dependent dimensional characteristics.