In an era of increasing digitization, the data we can collect and analyze has become increasingly complex. In particular we are wanting methods to make sense of data where each object itself has interesting structure or shape - for example we have a collection of graphs or a collection of scanned geometric objects. Topological Data Analysis provides methods to create a topological signature of each object that captures topological and geometric information.  Because these topological signatures lie in a common space, we can statistically analyse these topological signatures. This can be both more tractable than working with the raw data and also can identify relevant global and local features. 

One facet of my research is statistical theory for working with these topological signatures - and in particular statistical theory about persistence diagrams. Another facet is understanding the algebraic structures involved in persistent homology.

Important directions of research include developing topological tools for specific kinds of data structures such as geometric shapes and directed graphs and time varying systems. I was awarded a 2020 Discovery Early Career Research Award to research theory and applications of topological methods to perform statistical shape analysis. The main tools are the persistent homology transform and the Euler characteristic transform. The goal is to quantitatively compare geometric objects such as a set of bones, tumours, leaves, bird beaks, etc.