I have a passion for probability theory. My interest is both theoretical in nature, as well as aimed towards applications of probability theory in other areas of research. In particular, I combine probability theory with the fields of differential geometry and graph theory.
Random geometric graphs with degree constraints
Random graph models are used in a wide variety of applications to model and analyse the behaviour of networks. You can think of the Internet, or social networks, but also the construction of large molecules or polymers. Currently, I am working on random graphs with different types of constraints. On the one hand, I consider spatial constraints, which result in so called random geometric graphs. In such graphs, the vertices are random elements of some space which has a notion of distance. The distance between vertices is then taken into account when determining whether there exists an edge between these vertices.
The second type ofconstraint I am interested in are degree constraints. This means that every vertex can only form a prescribed number of connections. Such a random graph model is often referred to as the configuration model.
The main challenge I want to tackle is what happens when both the geometric and degree constraints have to be satisfied simultaneously. These constraints heavily interfere with each other, making existing methods for the two models seperately less viable to study these networks. Regarding these random geometric graphs with degree constraints, I am both interested in theoretical question, for instance about its structure, as well as practical questions about algorithms for sampling such networks.
Random walks and diffusions in manifolds
I am also generally interested in studying stochastic processes, such random walks and diffusions. In particular, I study such processes in Riemannian manifolds. My research mainly focusses on their asymptotic behaviour, in particular on large deviations. Large deviations are concerned with the limiting behaviour on the exponential scale of a sequence of random variables. Usually, this sequence of random variables satisfies a law of large numbers, and deviations are considered 'large' if it are deviations on this scale.
This work also extends to evolving Riemannian manifolds, in which the Riemannian metric is time-dependent. One can for instance think about inflating a balloon or for instance the evolution of a (biological) cell. A well known example of a time-dependent Riemannian metric is the Ricci-flow, which intuitively homogenizes the curvature of the manifold. However, I do not necessarily focus on specific examples, and the results are valid under some general regularity conditions of the time-dependence.
If any of the above topics interests you, don't hesitate to contact me at firstname.lastname@example.org. I am always happy to discuss and share ideas!