Chelsey Erway

Building on the pioneering work of Borel and Lebesgue, in the early 20thcentury Felix Hausdorff developed a concept of measure and dimensionthat has become indispensable to geometric measure theory. Hausdorffdimension generalizes our usual notion of dimension, allowing it to take onfractional values and giving us a way to describe sets that might otherwisebe problematic, most notably, self-similar sets.The talk will outline some basic properties of the Hausdorff measure andwill show how the Hausdorff measure allows us to assign fractionaldimension to some interesting self-similar sets such as the von Koch curve.The presentation will also describe the construction and properties of ahistorically important self-similar set: the space-filling curve first describedby Peano in 1890. No prior knowledge of measure theory required.