Research focus of the group is arithmetic properties (irrationality and transcendence) of the values of analytical functions, which usually come as solutions of differential or other functional equations. This includes arithmetic investigations of special integer sequences, in particular, automatic and regular sequences that find numerous applications in areas as diverse as group theory and cryptography. Another aspect of research is arithmetic applications of modular forms to the combinatorial problems of point count on algebraic varieties and connections of number theory to algebraic geometry. The group has strong collaboration links with researchers in Europe, Northern America and Asia, and is a leading Number Theory group within Australia.
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The Educational Research Group has interests spanning tertiary and secondary mathematics education, with strong threads relating to equity, enabling and excellence. Topics of interest to our group members include: inquiry learning, inquiring learning as benefiting equity and access, the relationship between formal and informal learning, computational and algorithm thinking, the use of ICT and more broadly of Tools in teaching and learning mathematics and statistics, experimental mathematics and visualisation as learning and engagement tools, the fitness for purpose of textbooks in learning, cognitive load theory, dispositions that promote learning, growth mindsets, problem and project based learning, authentic learning, learning communities and communities of practice.
Our members within the University typically sit within the School of Mathematics and Physical Sciences or the School of Education. We have close connections in the Centre for Teaching and Learning, the English Language and Foundations studies unit and the Centre of Excellence for Equity in Higher Education. We also have strong connections with Educational researchers in other Australian Universities and internationally.
A very large number of mathematical models from different areas of science can be written as differential equations. Finding the solutions to these equations is challenging and requires multiple stranded methods to obtain qualitative and quantitative properties. This group has a wide range of experience in numerical solution techniques and in their application in areas ranging from geophysics to control theory. We also have specialized knowledge of the finite element method, which has proved to be the most versatile and effective way to solve partial differential equations numerically.
The research in the group covers a range of combinatorics and discrete mathematics, from both pure and applied points of view.
Graphs and networks are a unifying theme for large parts of the group, relating to which members contribute to multiple areas of graph theory such as graph decompositions, graph labelings and graph algorithms. There is also an interest in enumerative combinatorics for various combinatorial objects including some arising out of mathematical physics. Further, structural and algorithmic questions relating to combinatorial objects, such as Hadamard matrices and antichains in partially ordered sets are studied. This has applications in areas as diverse as coding theory, data security and privacy.
A particular research strength is combinatorial optimisation, typically using methods from mixed integer programming and applied probability. This includes contributions to the theory of mathematical optimisation and applications in operations reseacrh and diverse supply chain contexts with a variety of industry partners from the local region and beyond.
The group has a strong national and international collaboration network.