Controlling the behaviour of eigenvalues: The interlacing method

This thesis uses the method of interlacing polynomials to study the behaviour of eigenvalues of a matrix after a rank-one update. Specifically, interlacing polynomials, common interlacing and interlacing families are exhaustively studied. These are excellent tools to find bounds on the eigenvalues of updated matrices by keeping track of how they...

Finding Structures within Large Point Sets

We investigate two open problems in discrete geometry regarding how large subsets of sets of points need to be in order for certain structures to emerge. First of all there is the Erdős-Szekeres convex polygon problem, also known as the Happy Ending problem. Interestingly, there is a clear distinction between the number of points required to...

Optimal Strategies of Autonomous Reconnaissance Missions

The role of Unmanned Aerial Vehicles (UAVs), more commonly known as drones, in society continues to become more significant every day, both in everyday life and in military operations. The extent to which unmanned vehicles are used for both offensive as well as reconnaissance missions is at an all-time high. To expand the number of operational...