In a world of abundant information, efficiently understanding the significance and structure of data is of paramount importance. When little is known about the source of our data points, it is often reasonable to assume that they are randomly distributed over the space of possible outcomes. The current project aims to investigate several structural properties of point sets distributed randomly over a geometric space. It is divided into two parts.

The first part of the project focuses on optimisation problems concerning random point sets and their statistical and probabilistic properties. In particular, we aim to make progress on questions such as:

- How uniformly is the set of random points distributed?

- Do large regions of a given shape (for example, rectangles) avoiding the point cloud exist?

- Can one efficiently explore the geometric point set algorithmically?

- Is it possible to separate the sampled points using simple geometric objects such as straight lines or planes?

The second part explores the structure of random geometric graphs, which are classical objects in the probabilistic theory of random graphs. A random geometric graph is defined on a random point set by connecting two points whose distance is smaller than a given positive real number. We aim to investigate aspects of the rich structure of random geometric graphs, including the symmetries they exhibit, the possibility of decomposing them into copies of a smaller graph, and their robustness under adversarial modifications.