Random Discrete Structures, Matrices and Topology

​Randomness in combinatorial structures, particularly random matrices, is a fascinating study area combining probability theory with combinatorics. Random matrix theory examines the distribution of eigenvalues in matrices with random entries, providing insights into complex systems such as nuclear energy levels and data covariance. It's instrumental in fields like quantum physics and finance, where the eigenvalue spectra reveal underlying system properties. Combinatorial structures, such as random graphs, employ randomness to explore network connectivity and robustness, impacting algorithms and social network analysis. For higher dimensional structures, stochastic topology studies random topological spaces, analysing structural changes under random conditions with applications to understanding the shape of complex biological forms and the topological properties of high-dimensional data spaces.

Understanding phase transitions random systems is an active area of research in the centre

Random trees are an example of important models of study