Algebra, Number Theory & Representations
Algebra is the study of symmetries and structures that model symmetries. Number theory is the study of the integers and number systems that have properties like the integers. These come together in Representation theory, which is the art of representing algebraic structures as matrices.
Mathematical analysis is a very broad area of mathematics, with strong connections both with other branches of mathematics, such as geometry and mathematical physics, and with other scientific disciplines, such as biology, chemistry, material sciences and finance. Roughly speaking, mathematical analysis focuses on the investigation of qualitative and quantitative properties of mathematical ``objects'' (e.g. functions, series, measures, spaces, solutions of differential equations, etc.). The techniques involved comprise those arising in the elementary calculus, such as limits, differentiation and integration, as well as sophisticated tools from functional analysis, harmonic analysis, complex analysis, differential geometry and geometric measure theory.
The Applied Mathematics Group has broad interests across the fields of colloid science, medicine, chemical engineering and materials processing. We often work on problems regarding the transport of materials, cells or molecules. Many of these problems arise from interaction with experimentalists, engineers, and industry partners, such as found in the Particulate Fluids Processing Centre, a Special Research Centre funded by the Australian Research Council, 2001-2005, or Royal Childrens Hospital. We develop discrete and continuum models to help understand these systems and generate and test new hypotheses. Particular strengths are in the mechanics of granular media , contact mechanics of deformable interfaces such as drops, the modelling of moving fronts of cells, random walks in random environments, and the mechanics of the atomic force microscope.
Complex systems play a key role in a vast range of societal activities– climate, the internet, traffic control, power distribution, agriculture, defence, manufacturing, engineering, water management, finance and many more. In any system, be it physical, biological or social, collective phenomena occur as the number of components increase. Analysing the behaviour of any individual component gives no indication as to how the system as a whole behaves, but understanding entire systems can lead to the prediction and subsequently the control and optimisation of their behaviour.
Discrete Mathematics & Algebraic Combinatorics
Discrete mathematics is the study of mathematical structures that are by nature discrete rather than continuous. It includes combinatorics and graph theory
Geometry & Topology
To understand a basic difference between topology and geometry, imagine a circle. Now ask yourself: Is it a perfect circle? If so, what radius is it? If you cannot immediately answer these questions, your concept of a circle is topological, to do with form rather than precise rigid/geometric shape. A topological circle can manifest geometrically in many different ways ... as an almost perfect circle (as drawn by someone with a steady hand), as ellipses of different shape and size, as pieces of string with ends tied together ... But another thing is very clear: a "circle" is different from a line or an interval. In a similar way, the 2-dimensional surfaces of a ball or donut are intrinsically different, no matter how you try to stretch or distort. Each in its own way nonetheless has very nice possible geometric shapes: a sphere may manifest as a soap bubble wobbling in the air, but also tries to form a perfectly uniformly curved surface: a "perfect" sphere. On the other hand, if you slice an actual donut into two pieces with a knife, several possibilities arise for the general "shape" of the two pieces ... they may both be rings, may both be like bent balls, one bit may be a ball and the other like the original donut ... all depending on how the cut was made. There are only finitely many topological possibilities, but infinitely many geometric ones. The research group at Melbourne studies the interplay of geometry and topology, as well as some applications to processes in industry, and models of biological systems. For example, DNA can form closed loops, and there are serious difficulties understanding or modelling enzyme actions which permit the unlinking of two copies after replication. Recent work in mathematical physics of elementary particles and statistical mechanics has provided some ways to try to understand this phenomena. Related mathematics also occurs when considering the "perfect" geometric forms of possible 3-dimensional topological spaces. The most important -- hyperbolic geometry -- is intimately connected with geometric transformations which occur in the theory of relativity. When equations are used to describe things, such as the relationships of different lengths of components of a mechanical robot, the geometry and topology of possible shapes is again important to understand and describe: Fixing a length is akin to slicing a donut as described above. Problems related to combinatorial optimization often appear, which commonly are of great interest in industry. Being able to describe, display and compute such objects is also an active research area in this group.
Learning and Teaching Innovation
Innovation in the teaching of mathematics and statistics is a key focus of the department. This group fosters innovations in learning and teaching for tertiary mathematics and statistics.
Mathematical and Computational Biology
Mathematical, statistical and computational methods are crucial in many areas of modern biological research. Conversely technological advances in biology allow more data, often of a novel type or at a finer resolution, to be collected resulting in new challenges that are motivating research in mathematics, statistics and computational methods.
Mathematical Physics & Statistical Mechanics
Mathematical Physics is the study of the mathematics associated with models of the physical world. One important and modern part of mathematical physics is the study of models in statistical mechanics. Statistical mechanics involves the understanding of large complex systems by averaging the behaviour of the individual components. For example, one can understand the behaviour of a gas without describing the motion of all the molecules involved, simply by knowing the type and strength of the forces between the molecules, using the principles of statistical mechanics. This powerful idea can be applied to many and varied systems in the natural world and in the human arena. It was not said lightly by a leading scientist that 'a well-trained statistical mechanician can tackle any problem' since, for example, statistical mechanics graduates can be found working in high-end financial organisations, in brain research or working for the Human Genome Project.
The discipline of Operations Research (OR) provides a scientific approach to decision making. Also known by titles such as Management Science, or Logistics, Operations Research involves formulating mathematical models of decision making problems, and developing or applying mathematical tools to obtain solutions. Many businesses and all large complex organisations face difficult decisions on a daily basis. For example, a manufacturing company must decide how much of each product it should be making at each point in time, how many products of each type it should keep in inventory, by what modes of transport and what routes it should distribute its product, which combination of new product development projects it should fund in the next year, which workers should be rostered on which shifts, how much overtime will be required of each worker, when it should replace or repair its equipment, and so on. The decisions made interact with each other and may have complex repercussions that are difficult to evaluate. Each decision involves making a trade-off between competing activities, often vying for limited resources. For example, a decision to maintain steady production levels may reduce production costs but increase inventory costs, while a decision to produce a large quantity of one product may deplete stock of a component needed for the timely production of another product. The mathematical techniques used in Operations Research are drawn from areas of mathematics such as Optimisation, Optimal Control and Probability and Statistics. The area of Optimisation, which is concerned with the mathematical problem of minimising or maximising a function or functions subject to constraints, plays a particularly important role since the objective of many decision making problems is to determine the minimum or maximum out of a set of alternatives; for example a production company might wish to make decisions so as to minimise its transportation costs or to maximise its workers' job satisfaction. Major subfields of Optimisation which are critical in solving Operations Research problems are Mathematical Programming, Dynamic Programming, Network Optimisation and Stochastic Modelling, all of which are the subject of active research by members and graduate students of the Operations Research Group in the Department of Mathematics and Statistics.
The Statistics group is interested in the application of statistical theory in a rich variety of contexts. The research interests of the group include biostatistical issues such as meta-analysis and survival analysis, food science statistics such as measuring and describing the quality of consumables, environmental applications including population modelling, group testing, drug testing in sport, bioinformatics and sample surveys. The group's link with the Statistical Consulting Centre means that there are research opportunities arising from real-world applications that are readily available.
Probability is a beautiful and ubiquitous field of modern mathematics that can be loosely described as the mathematics of uncertainty. It has applications in all areas of pure and applied science, and provides the theoretical basis for statistics. Four of the last twelve Fields Medallists have been recognised for their work in probability.
Stochastic processes involves the study of systems that evolve randomly in time. The latter is a characteristic feature of the behaviour of most complex systems, for example, living organisms, large populations of individuals of some kind (molecules, cells, stars or even students), financial markets, systems of seismic faults etc. Being able to understand and predict the future behaviour of such systems is of critical importance, and this requires understanding the laws according to which the systems evolve in time. Discovering such laws and devising methods for using them in various applications in physics, biology, statistics, financial engineering, risk analysis and control is the principle task of researchers working in the area of stochastic processes. Computer simulations also play an important role in the field, and enable one to get insight into the behaviour of analytically intractable systems.
Research in our group covers a diverse range of theoretical and applied probability and stochastic processes, including: stochastic approximation, the theory of queues and stochastic networks, random walks, random graphs and combinatorial structures, reinforcement processes, interacting particle systems, stochastic dynamical systems, boundary crossing problems, and applications in epidemiology, healthcare, traffic management, risk modelling, financial engineering.
Students interested in pursuing a career in various fields such as mathematics, statistics, physics, biology, finance, economics etc. will benefit greatly by studying probability at a deep level. Stochastic Processes graduates work in research and development departments of leading financial and insurance institutions, defence organisations, as well as in the areas of bioinformatics, signal processing, technology and many others.