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 organizations 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 Optimization, Optimal Control and Probability and Statistics. The area of Optimization, which is concerned with the mathematical problem of minimizing or maximizing 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 minimize its transportation costs or to maximize its workers' job satisfaction. Major subfields of Optimization which are critical in solving Operations Research problems are Mathematical Programming, Dynamic Programming, Network Optimization 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.
The Stochastic Processes Group in the Department works in the area of probability theory - a branch of mathematics providing means for modeling uncertainty. The latter is a characteristic feature of the behaviour of most complex systems - e.g. 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 knowing the laws according to which the systems evolve in time. Discovering such laws and devising methods for using them in various applications including statistics, financial engineering, risk analysis and control is the principle task of researchers working in the area of Stochastic Processes. The results of the theory turn out to be useful not only in applications to random phenomena, but also in some areas of pure and applied mathematics. Mathematically the theory of stochastic processes is a very challenging and still actively developing theory. It requires deep knowledge of different areas of mathematics, including classical calculus, functional analysis, measure and function theory, algebra and combinatorics. Computer simulations also play nowadays an ever increasing role and enable one to get insight into the behaviour of analytically yet intractable systems. Stochastic Processes graduates work in research and development departments of leading financial and insurance institutions, defence organizations, as well as in the areas of bio-informatics, signal processing and many others. Since they usually would have a concurrent training in statistics, our graduates are also highly employable in a huge variety of areas requiring specialists in statistics. Research in the Stochastic Processes Group is focused on a number of areas, ranging from more theoretical ones such as point processes approximation, the theory of stochastic networks and boundary crossing problems to applications of stochastic processes to risk modelling and financial engineering. Our group currently consists of Kostya Borovkov and Aihua Xia. They both are members of the newly established ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS). It is worth noting that a student holding an APA and undertaking a PhD under the supervision of a MASCOS member will be considered for a top-up scholarship plus an additional grant for research-related expenses.