Discrete mathematics

The research of the members of the discrete mathematics group is very versatile and covers several aspects of this mathematical area. The group also has strong connections to a number of overseas collaborators. Prof Helmut Prodinger, Dr Dimbinaina Ralaivaosaona and Prof Stephan Wagner share a particular interest in enumerative, analytic and probabilistic combinatorics. This involves combinatorial counting problems (exact and asymptotic enumeration) and the probabilistic analysis of a variety of discrete structures, such as strings, compositions, partitions, and trees. Typical research problems that are investigated in this context involve parameters of discrete structures (for instance the height of a tree) whose distributional properties are of interest: mean values, variances, and the distribution in the limit when the investigated structures grow very large. This type of question is for example of major interest in the analysis of algorithms, where parameters of data structures translate to the performance of algorithms. Other areas in which questions of a similar nature arise include physics (especially statistical physics) and bioinformatics (for instance in phylogenetics). Prof Prodinger also has a strong interest in the study of identities of a combinatorial nature, involving number sequences that play a role in combinatorics, such as (q-)binomial coefficients and Fibonacci numbers. Moreover, he works on non-standard (redundant) digit representations, motivated by performance questions in cryptography. Graphs, in particular types of random graphs, which can be used to model networks of different kinds, play a major role in this field of research. Certain aspects of pure graph theory, in particular extremal questions, are also a main interest of Prof Stephan Wagner; the research problems, which arise for example in theoretical chemistry, usually involve an invariant (such as the average distance between vertices) whose value is to be maximised or minimised in a prescribed class of graphs. Dr Karin-Therese Howell’s research interest in graph theory is in the hereditary properties of graphs and in using graphs to study algebraic structures. Prof Marcel Wild’s research covers yet another aspect of discrete mathematics. In his research on Boolean functions, he is using wildcards for the purpose of compression of their model set. This also has applications to discrete algorithms and involves programming in Mathematica. Some of his high-level Mathematica programmes best corresponding hardwired commands like BooleanConvert. Moreover, he has an interest in lattice theory, in particular modular lattices. A recent research project he has been working on concerns the embedding of modular lattices into partition lattices.