Ever wonder how one can do calculus on the Sierpinski gasket? How can one define a derivative or even a Laplacian on a fractal set? And once one has a Laplacian, can we study differential equations which may tell us how heat dissipates on metals with fractal molecular structure? Or how water may move on a dish with a fractal boundary?

Noncommutative fractal geometry is the study of fractal geometry and analysis on fractals, with operator algebraic tools. My work in this area focuses on how one can use spectral triples (C* algebra, a Hilbert space, and a Dirac operators) to formulate definitions of classic fractal geometric ideas such as dimension, metric, and measure.

I have a group of graduate students currently exploring how one can use fractal dimension to quantify how much chaos exists in a data set. Data sets arrising from EEG machines are partiicularly interesting for this purpose. Known notions for assigning a fractal dimension to such data include Higuchi's dimension and Katz dimension.