## Research

Mathematical sciences are the most fundamental and pervasive disciplines in human science and culture. The Centre for Research in Mathematics covers a continuum from very applied to very theoretical research, structured around the core themes of Algebra and its applications, and Analytical and statistical methods in control theory and biological systems.

### Theme 1: Algebra and applications

#### Group theory and combinatorics

This project deals with fundamental questions in group theory and combinatorics. Specific areas of interest are braid groups and combinatorial or geometric generalisations (Artin groups and Garside groups; braid-related monoids), graphs, and combinatorial designs. The members of this project are interested in both theoretical questions about the abstract structure of these algebraic objects, and algorithmic questions concerning efficient computations with these objects. Some members develop and use computational algebra tools to support their theoretical research.

**Key people:** James East, Andrew Francis, Volker Gebhardt

**Grants:** Algebraic evolution and evolutionary algebra, Andrew Francis

**Flyer:** Braids and Garside theory

#### Diagram algebras and semigroups

Starting from a diagram (think of cities and flights between them) one can generate an algebra that captures the movement along the diagram. In return, out of these algebras arise so called semigroups. Understanding the relations between the shape of diagrams and their associated algebras and semigroups have attracted the curiosity of mathematicians for over half a century. Grading (measuring the distance travelled along the diagram) appears quite naturally in the study of algebras generated from diagrams. In fact, the graded structure encodes a significant amount of information about the algebra and its associated diagram. All the indications of previous work are that taking into account the grading, one can completely understand or classify algebras that stem from diagrams. One part of the project is to focus on this approach.

One may also consider algebras whose objects are themselves diagrams. These algebras have strong connections to geometric objects such as knots and links and reflection groups. They are also of fundamental importance in many branches of semigroup theory as they contain the full transformation semigroups and the symmetric and dual symmetric inverse monoids. This project studies the above algebras and applications, as well as their connections with other areas of mathematics.

**Key people: **James East, Volker Gebhardt, Roozbeh Hazrat

**Grants: **Algorithmic approaches to braids and their generalisations, Volker Gebhardt

#### Algebraic biology

In this project evolutionary processes in bacteria are modelled by the action of a croup (such as a set of symmetries) on the space of bacterial genomes. The fundamental biological questions are how closely related different genomes are, and more deeply, what the relationships are among a family of genomes. This is essentially a question of determining phylogeny. The group theoretic questions that arise include how to find minimal expressions for a group element in terms of biologically reasonable generators, finding minimal spanning trees in a Cayley graph, and finding algebraic structures that model a range of biological processes simultaneously. The work involves proving results about the groups in question, and devising computational algorithms to solve problems using the group theory.

**Key people: **Andrew Francis, Volker Gebhardt

**Flyer:** Algebraic Biology

### Theme 2: Analytical and statistical methods in control theory and biological systems

#### Control theory and signal processing

Control theory investigates the fundamental behaviours of dynamical systems and related applications. This project addresses a number of research questions regarding complex dynamical systems including: the stability analysis of time-delay and singularly perturbed impulsive systems; consensus strategies for multi-agent systems; and new methods for system identification. In signal processing, the use of wavelets for de-noising Poisson and other non-Gaussian signals and images, as well as the use of spatial context in hyperspectral unmixing, are key topics of interest. These have wide applications in communications, image processing and data analysis.

**Key people:** Glenn Stone, Wei Xing Zheng

**Grants:** Quantized identification of feedback control systems, Wei Zheng**Grant: **Development of Identification Methods for Nonlinear Dynamical Systems, Wei Zheng

#### Mathematical and statistical modelling

Food science – methods to make better use of data from plate counting experiments. Plate counting experiments are used to determine the concentration of viable spores in a sample. Present data analysis imposes restrictions that result in up to two thirds of the data being ignored. We are developing methods to make better use of this data. (With Belinda Chapman, UTS, and Janelle Brown and Michele Bull, CSIRO.) Food science – flow cytometry. Plate counting is a laborious process and it is hoped that flow cytometry can provide the same information. There are significant data analysis challenges however, and we are addressing these. Biology. Research in this domain includes modelling evolutionary processes and statistical data analysis.

**Key people: **Andrew Francis, Glenn Stone