Deterministic world views are often contrasted with random or chance processes. Paradoxically, in mathematics, deterministic and probabilistic (or stochastic) descriptions are often two sides of the same coin. A well known example is the stochastic process of microscopic Brownian motion and its deterministic description in terms of the macroscopic diffusion equation (a partial differential equation).
Stochastic modelling is used in the study of many natural phenomena. One branch of research in our department takes its motivation from statistical physics, which studies the behaviour of systems of so many particles, that only a statistical description is feasible. Examples are ferromagnetic materials, modelled by the Ising model, and porous media, modelled by percolation theory. These systems undergo drastic changes in their macroscopic spatial properties as an external parameter is varied: ferromagnets lose their magnetization when the temperature becomes too high, and in percolation, we observe a transition from a state with only finite open clusters, to a state with a unique infinite open cluster as the density of open cells increases. We are especially interested in the spatial properties of such systems at and around the transition point. The study of these properties involves an exciting mixture of mathematics, using ideas from complex analysis, probability theory and stochastic processes.
Stochastics is also used to model the dynamics of natural phenomena. We study stochastic models for the spread of forest fires and infections, the Bak-Sneppen model of evolution, and sandpile models. Mathematical challenges in this area include finding the stationary states of these systems, and understanding the phase changes and critical behaviour they exhibit.
Similar questions arise in the study of dynamical systems. Although in these systems the time evolution of a point in a geometric space is given by a fixed, deterministic rule, they may generate beautiful and highly complex patterns. These patterns are often difficult to capture analytically. We use topological tools in conjunction with computational methods to study qualitative properties of the dynamics, which showcases the connections with other branches of mathematics.
The dynamics of deterministic systems are often chaotic and hard to predict, and a description in terms of invariant measures and ergodic properties becomes the best one can get. This illustrates that deterministic and probabilistic, statistical descriptions can go hand in hand. In fact, models with both deterministic and random components are becoming more and more important. Examples include stochastic differential equations (much used in mathematical finance, but also in neuroscience, for example) and dynamical systems with noise.