In the first lecture, we will provide an overview of the various ways that topological information is used in signal detection problems in functional MRI (fMRI) and other imaging applications. The principal tool used involves computing the expected number of critical points of various types of a smooth random field under some predetermined null hypothesis. We will describe roughly how some of these calculations can be carried out using the so-called Gaussian Kinematic Formula. In the second lecture, we will describe some typical dimension reduction tools used in statistics and machine learning. Not surprisingly, many of these techniques build on the SVD of some data-matrix. Topics covered will include (generalized) PCA, sparse PCA, some ICA and, time permitting, matrix completion.