The word stochastic refers to things that involve chance or probability. In mathematics, stochastic geometry involves the study of geometric objects that are determined by random variables. That is, stochastic geometry studies random shapes selected from a probability distribution.
My work involves random fields and random cubical complexes. A random field is basically a function whose value at each point is a random variable. I have studied Gaussian-related random fields, obtaining formulas for the expected values of the Hadwiger integrals of such fields and a connection to Hadwiger's theorem for functions. A random cubical complex consists of unit cubes with vertices on an integer lattice, chosen independently with equal probability. Working with Michael Werman, I have found polynomial formulas for the mean and variance of the intrinsic volumes of random cubical complexes, as well as a central limit theorem. I am now working to generalize these results.
The motivation for my work in stochastic gemometry is to understand noise in data. Stochastic processes can provide good models of the noise that often occurs in real-world data. By understanding the contribution of noise, we can more easily control for it when interpreting data.