School of Mathematics

Discussions about constructive mathematics are usually derailed by philosophical opinions and meta-mathematics. But how does it actually feel to do constructive mathematics? A famous mathematician wrote that "taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists." Was he right? In this talk we shall visit the astounding worlds of constructive mathematics.

A fundamental theorem in linear algebra is that any real n x d matrix has a singular value decomposition (SVD). Several important numerical linear algebra problems can be solved efficiently once the SVD of an input matrix is computed: e.g. least squares regression, low rank approximation, and computing preconditioners, just to name a few. Unfortunately in many modern big data applications the input matrix is very large, so that computing the SVD is computationally expensive.

Persistent homology is a central object of study in applied topology. It offers a flexible framework for defining invariants, called barcodes, of point cloud data and of real valued functions. Many of the key results of the last several years in the theory of persistent homology have been formulated in terms of a metric on barcodes called the bottleneck distance. There is a multi-parameter generalization of persistent homology, called multi-dimensional persistent homology, which is naturally suited to the study of noisy point cloud data.