Mathematics and statistics research
The research behind tomorrow’s breakthroughs
,
We are the only Australian university to have received the highest rating (5 out of 5) for research in the mathematical sciences in every Australian Research Council Excellence in Research for Australia assessment to date.
The interaction of algebra with diverse areas within and outside of mathematics leads to the formulation and discovery of new algebraic structures. Other structure theories analysed and employed by the group at the University of Sydney in order to uncover secrets about the formation and role of structure in a broad range of phenomena include dynamical and integrable systems, networks and complex systems.
Key researchers:Kevin Coulembier,Zsuzsanna Dancso,Gus Lehrer,Andrew Mathas,Alexander Molev,James Parkinson,Anne Thomas,Geordie Williamson,Oded Yacobi,Ruibin Zhang.
Symmetry is everywhere, we see it in the geometry of everyday objects, but also in differential equations or even in the laws of physics. Lie theory is the mathematical framework for understanding and using these symmetries.
One of the central problems in Lie theory is how to ‘represent’ symmetries, leading to representation theory. This theory is an important gateway for applications of Lie theory to areas like coding theory and quantum chemistry. A classical example is the representation theory of the rotation group dictating the energy level structure of the hydrogen atom.
Our new results are often obtained by establishing original connections with other fields of mathematics. For instance, we derive and study:
Seminars
Key researchers:John Cannon,Kevin Coulembier,David Easdown,Alexander Fish,,Nalini Joshi,Gus Lehrer,Andrew Mathas,James Parkinson,Bregje Pauwels,Jonathan Spreer,Anne Thomas,Stephan Tillmann,Geordie Williamson,Oded Yacobi,Ruibin Zhang, John Voight.
Group theory is central to modern mathematics and its applications. The relationships between solutions to equations, positions of an object in motion, atoms in a crystal or letters in a genetic code are all described by a group, and group theory is essential to encryption.
Many other fundamental abstract structures in algebra, such as rings, fields and vector spaces, are generalisations of groups.
Research on group theory and related concepts at the University of Sydney includes both computational algebra and the investigation of theoretical questions for their own sake. Our work connects closely to algebraic geometry, representation theory, integrable systems, ergodic theory, geometric group theory and low-dimensional topology.
Seminars
Software
is a large, well-supported software package designed for computations in algebra, number theory, algebraic geometry, and algebraic combinatorics.
Key researchers:Dmitry Badziahin,John Cannon,Alexander Fish,Nicole Sutherland,Geordie Williamson, John Voight.
Despite their apparent simplicity, numbers display an incredible richness. Number theory is an incredibly broad area of mathematics, and the main problems of interest investigated at the University of Sydney include:
Approximating "complicated" numbers
For example, Diophantine approximation aims to approximate real numbers with rational numbers as precisely as possible. Applications include signal processing and cryptography.
Fast computational implementation of operations
While operations like the multiplication of numbers on a calculator appear to be instant, other tasks require millions of operations on numbers with millions of digits. This could take days or even weeks.
Making these calculations faster is a goal of computational number theory. It is especially useful in cryptography, which specifically selects operations that are known (or suspected) to be computationally difficult.
Sum-product phenomena
Looking at the multiplication and addition tables of numbers, we observe that the total number of different values in the former one is much larger than in the latter. This is an instance of the sum-product phenomenon, an area of study in additive combinatorics. It is applied in theoretical computer science, cryptography, and other areas of pure mathematics.
Key researchers:John Cannon,Kevin Coulembier,Zsuzsanna Dancso,,Gus Lehrer,Andrew Mathas,Alexander Molev,James Parkinson,Bregje Pauwels,Anne Thomas,Kane Douglas Townsend,Geoff Vasil,Geordie Williamson,Oded Yacobi,Ruibin Zhang, John Voight.
Representation theory is a cornerstone of modern mathematics, as it allows us to describe symmetries of objects that are not manifestly geometric. It plays a crucial role in number theoryand has many important applications within the mathematics and the sciences, including in particle physics, cryptography, and quantum computation.
The University of Sydney employs a world-class group of researchers in representation theory, working in various fundamental areas. These include:
Representations of the symmetric group and related algebras
In particular, our researchers focus on representations of the symmetric group over fields of positive characteristic, which are not well-understood.
Geometric representation theory
This theory applies algo-geometric tools to Lie theory, the mathematical framework of symmetries.
Categorical theory and representation theory
These areas study categories of representations from both the Tannakian perspective (reconstructing the group from its representations), and higher representation theory (studying symmetries of the categories themselves).
Invariant theory
One of the oldest branches of representation theory, it studies the effect of a group acting on polynomial functions on a space or variety.
Quantum groups
Originating in statistical mechanics, quantum groups are deformations of universal enveloping algebras of lie algebras, or closely related algebras.
Key researchers:Eduardo Altmann,Clio Cresswell,Georg Gottwald,Nalini Joshi,Rachel Wang.
Complex systems are abound in the natural sciences, engineering and social sciences. They involve dynamics that occurs on vast temporal and spatial scales; from Earth’s global climate, the human brain, to software systems and cities.
Network theory provides a powerful language to eke out the essential information needed to understand and control complex systems.
Given their sheer complexity, a detailed description of complex systems may not be possible, even with advanced computational tools. Mathematicians address this challenge by identifying relevant macroscopic variables and describing their dynamics or mutual relationships. Both are highly non-trivial tasks that require sophisticated mathematics.
In our research, we develop novel methodologies to improve our understanding of complex systems, with applications to gene and brain networks, language evolution, natural language processing, information spreading in social media, coupled oscillators and climate dynamics.
The task of unravelling qualitative statistical behaviour of complex systems and functional structure in networks requires us to borrow theory from dynamical systems theory, graph theory, statistical mechanics, inverse modelling and Bayesian statistics, to name but a few.
Key researchers:Zsuzsanna Dancso,Alexander Fish,Andrew Mathas,Alexander Molev,James Parkinson,Jonathan Spreer,Anne Thomas,Geordie Williamson,Oded Yacobi.
Counting problems arise all throughout mathematics and science: in optimisation, algebra, probability theory, topology and geometry, computer science and statistical physics. Combinatorics is the science of counting smart, and it encompasses the related study of finite and discrete structures that arise in a wide variety of contexts.
Questions of counting combinations and permutations were recorded as early as the 6thcentury BC, and increasingly complex discrete problems emerged throughout history. Powerful theoretical frameworks for counting and working with combinatorial structures were developed in the second half of the 20thcentury, raising the profile of combinatorics to a prominent field in its own right.
The study of large networks—such as the Internet, social networks, and the brain—has led to a recent explosion of progress in combinatorics, employing techniques from many fields of mathematics and statistics, as well as advances in Artificial Intelligence (AI) and computing.
Our researchers investigate the multi-faceted interplay between combinatorics and algebra, topology, geometry, dynamics and AI.
Key researchers:Emma Carberry,Harini Desiraju,Holger Dullin,Nalini Joshi,Robert Marangell,Jae Min Lee,Milena Radnovic,Pieter Roffelsen.
Integrable systems form the core of classical mechanics, modern mathematical physics and special function theory, arising in applications that are widespread and growing rapidly. Examples include plasma physics, elementary particles, superconductivity, and non-linear optics.
One area of intense recent activity in which integrable systems arise is the study of energy levels of heavy particles in atomic physics, which is closely connected to the spectral properties of random matrices.
This serendipitous connection has given rise to some of the most active and fruitful developments of mathematics in recent times.
Applications of random matrix theory are numerous and far-reaching: from particle physics to the distribution of airline boarding times andthe zeros of the Riemann-zeta function along the critical line.
Our research interests lie in:
Key researchers:Eduardo Altmann,Dzmitry Badziahin,Chris Bertram,Nathan Duignan,Holger Dullin,Alexander Fish,Georg Gottwald,Nalini Joshi,Robert Marangell,James Parkinson,Milena Radnovic,Anne Thomas,Stephan Tillmann,Martin Wechselberger,Geordie Williamson.
Dynamics is the study of the limiting behaviour of an observable variable, such as temperature or pressure, along with time evolution. Dynamical systems are manifolds or other topological spaces endowed with a group action.
One of the first examples of a dynamical system was introduced by Newton, who studied the trajectories of planets in the solar system. While trying to prove the stability of the solar system in 1890, Poincaré initiated qualitative methods to study dynamical systems preserving volume. These insights became a cornerstone of ergodic theory.
At the University of Sydney, research on dynamical systems and ergodic theory combines the study of theoretical questions and the application of dynamical methods. Theoretical areas of dynamics includebilliards, chaos, ergodic theory and integrable systems.
Our researchers apply dynamical methods in biology, climate, complex networks,machine learning,physiology, as well as many other fields. Using dynamics, we also provide insights into other areas of pure mathematics such as additive combinatorics, algebra, geometric topology, geometric group theory and number theory.
Seminars
Key researchers:
Geoff Bailey,John Cannon,Allan Steel,,Don Taylor,Bill Unger,JohnVoight,
Computers allow us to explore and solve hard problems in pure mathematics. In computational algebra, we study a wide range of topics--including algebra, number theory, geometry, representation theory, and combinatorics--through the lens of symbolic computation and with an eye to explicit structures.
Using sophisticated algorithms, we can manipulate complex mathematical objects, provide examples or counterexamples, and verify statements that would be otherwise challenging or impossible to handle.
Our work is focused on the development of,a large, well-supported software system designed for algebraic computation.
Magma was first released in 1993 at the University of Sydney and has received contributions from hundreds of mathematicians worldwide.
Magma provides a mathematically rigorous environment, a vast library of algorithms, and a flexible language for defining and working with many structures in pure mathematics, including groups, rings, fields, modules, algebras, schemes, curves, graphs, designs, codes, and more.
Our work pursues several lines of investigation, including basic research in pure mathematics to improve general understanding of Analysis and Probability and their interactions, as well as development of mathematical results that lay the groundwork for real-world applications in engineering, social and natural sciences, medicine and healthcare.
Key researchers:Florica Cirstea,Daniel Daners,Ben Goldys,Georg Gottwald,Daniel Hauer,Nalini Joshi,Robert Marangell,Martin Wechselberger,Haotian Wu,Zhou Zhang.
Partial differential equations (PDEs) play a key role in modelling real-world phenomena occurring in physics, chemistry, biology, and economics. In a given model, PDEs represent the mathematical description of different laws in a system interacting with each other.
Within the fundamental goal of solving PDEs of different types (elliptic, parabolic, or hyperbolic), mathematical questions that arise include determining the existence, uniqueness, and qualitative properties of solutions.
To solve these problems, our research group contributes with new results by investigating:
To achieve our results, we often exploit geometric (e.g. concavity/convexity) properties of solutions, apply maximum principles, abstract functional analytic concepts such as linear and nonlinear semigroup theory, and use concepts from stochastic analysis and optimal transport theory. This often leads to the development of new analytical methods.
Seminars
Key researchers:Jennifer Chan,Clara Grazian,Uri Keich,Linh Nghiem,John Ormerod,Shelton Peiris,Michael Stewart,Qiuzhuang Sun,Garth Tarr,Qiying Wang,Rachel Wang.
Statistical methods are the bedrock upon which quantitative science is built. These in turn are built upon probabilistic statistical models, which determine which methods are appropriate and/or valid in different applied contexts.
Beyond the scientific sphere, understanding what is and is not possible for different statistical procedures is also crucial for society more broadly: a statistically literate population is the best weapon against questionable statistical claims (“lies, damned lies and statistics”).
Research into statistical theory defines the capabilities and limits of statistical methods and reasoning. Development of the statistical theory underlying today’s highly complex data models is a major challenge.
At the University of Sydney, we have made significant progress expanding the theory for models appropriate for classification, clustering (and other machine learning methods), dependence and extremes, high-dimensional inference, model selection, survival analysis and time series analysis.
These methods are in turn applicable to a vast range of applications: biological, econometric, educational, environmental, financial, industrial, marketing, medical, psychological, and many others.
Seminars
We investigate mathematical underpinnings of computing, with the goal of designing algorithms and computational approaches to problems in engineering, the social and natural sciences, medicine and healthcare. The overarching goal is to build a mathematically sound foundation for the design of modern computational thinking and to develop analytic frameworks for their correctness, performance, reliability and security.
Key researchers:Georg Gottwald,Lindon Roberts,Geordie Williamson,Dinxuan Zhou,Pengyi Yang.
In the past decade, machine learning has led to a paradigm shift in computer vision and language processing with remarkable success in, amongst others, medical image analysis and drug discovery.
Mathematics has an integral part to playand is in the unique position to unravel some of the mysteries of the success of machine learning algorithms, drawing from numerous areas within mathematics including numerical analysis, optimization, probability theory, group theory,approximation theory and dynamical systems.
We develop and apply machine learning algorithms to formulateconjectures in pure mathematics with the aim to prove them. We apply machine learning to understand molecular trans-regulatory networks in complex systems biology and develop new methods to forecast dynamical systems.
Key researchers:Eduardo Altmann,Jennifer Chan,Georg Gottwald,Clara Grazian,Uri Keich,John Ormerod,Ellis Patrick,Michael Stewart,Garth Tarr,Rachel Wang,Geordie Williamson,Jean Yang,Pengyi Yang.
The advent of Big Data has revolutionised science and industry, bringing new challenges and opportunities in the form of large-scale, heterogeneous, and dynamical data.
Much of the research in computational statistics and machine learning is inspired by practical problems in data-rich disciplines including bioinformatics, artificial intelligence, finance, and social science, where mathematical and algorithmic creativity are interwoven into statistical methodologies.
Our research spans a diverse spectrum from theory to applications. We tackle challenging theoretical problems concerning the mathematical and statistical foundations of machine learning frameworks and algorithms.
We develop innovative and efficient computational methods to extract insights from data, building predictive models and performing statistical inference.
Key researchers:Geoff Bailey,John Cannon,Allan Steel,,Don Taylor,Bill Unger,JohnVoight,
Computers allow us to explore and solve hard problems in pure mathematics. In computational algebra, we study a wide range of topics--including algebra, number theory, geometry, representation theory, and combinatorics--through the lens of symbolic computation and with an eye to explicit structures.
Using sophisticated algorithms, we can manipulate complex mathematical objects, provide examples or counterexamples, and verify statements that would be otherwise challenging or impossible to handle.
Our work is focused on the development of,a large, well-supported software system designed for algebraic computation.
Magma was first released in 1993 at the University of Sydney and has received contributions from hundreds of mathematicians worldwide.
Magma provides a mathematically rigorous environment, a vast library of algorithms, and a flexible language for defining and working with many structures in pure mathematics, including groups, rings, fields, modules, algebras, schemes, curves, graphs, designs, codes, and more.
The construction ofmathematical modelsin the sciences, medicine, technology and finance is an important and rapidly expanding area of research. Our researchers design efficient algorithms for processing large amounts of data, develop dynamical models to explain natural phenomena and discover new ways to detect and interpret complex patterns in real-world systems.
Key researchers:Georg Gottwald,Lindon Roberts,Geordie Williamson,Dinxuan Zhou,Pengyi Yang.
In the past decade, machine learning has led to a paradigm shift in computer vision and language processing with remarkable success in, amongst others, medical image analysis and drug discovery.
Mathematics has an integral part to playand is in the unique position to unravel some of the mysteries of the success of machine learning algorithms, drawing from numerous areas within mathematics including numerical analysis, optimization, probability theory, group theory,approximation theory and dynamical systems.
We develop and apply machine learning algorithms to formulateconjectures in pure mathematics with the aim to prove them. We apply machine learning to understand molecular trans-regulatory networks in complex systems biology and develop new methods to forecast dynamical systems.
Key researchers:Chris Bertram,Daniel Daners,Peter Kim,Robert Marangell,Mary Myerscough,Martin Wechselberger.
We use mathematics to explore fascinating and complex questions of living systems. To do so, we team up with collaborators from relevant disciplines, creatively build mathematical models, and seek to provide insights from mathematical reasoning.
Our mathematical biology group works in areas as diverse as anthropology, collective behaviour, ecology, epidemiology, immunology, medicine, neuroscience, physiology, social insects, and beyond.
These systems span across a vast breadth of spatial scales, from the microscale study of bacteria and viruses to the global scale of human populations and ecosystems. Timescales range from milliseconds for neural transmission to millions of years for evolutionary adaptation.
Our mathematical biologists use mathematical, statistical, and computational tools to develop, improve and solve mathematical models. We use differential equations, difference equations, and agent-based models to capture the dynamics of the world around and within us.
Key researchers:Jean Yang,John Ormerod,Uri Keich,Pengyi Yang,Ellis Patrick,Rachel Wang,Garth Tarr,Clara Grazian,Shila Ghazanfar.
Bioinformatics is an interdisciplinary field utilising quantitative methods from computer science, mathematics, and statistics to analyse and interpret large biological datasets. We develop and apply methods for applications ranging from the identification of predictive biomarkers of disease to characterising molecular signalling patterns within cells.
We share an interest in developing statistical and computational methodologies to tackle the foremost significant challenges posed by modern biology and medicine.
Most of the researchers in the School of Mathematics and Statistics working in Bioinformatics are also members of the Bioinformatics Cluster in theSydney Precision Data Science Centre.
Some of the software our group develop can be found on the public repository of.
Seminars
Key researchers:John Ormerod,Jennifer Chan,Clara Grazian,Rachel Wang.
Bayesian statistics is an approach to data analysis where epistemological uncertainty associated with model parameters is described using probability distributions. As data is gathered, Bayes Theorem is used to update the probability distribution of model parameters given the observed data.
Bayesian methodology is used in diverse fields such as machine learning, medicine, ecology, insurance, finance and astronomy, to name just a few.
Research in Bayesian statistics at the University of Sydney is at the forefront of the area. Our group conducts research in the Bayesian analysis of time series data, including stochastic volatility models, modelling of cryptocurrencies via multivariate time series, and modelling of COVID-19 case numbers.
We consider approximate Bayesian inference methods where numerical accuracy is sacrificed in order to enable methods for complex datasets arising in machine learning and bioinformatics.
Finally, we consider problems involving various forms ofclustering (including mixture modelling), nonparametric Bayesian methods, and community detection methods on graphs.
Key researchers:Kevin Coulembier,Zsuzsanna Dancso,Gus Lehrer,Andrew Mathas,Alexander Molev,James Parkinson,Anne Thomas,Geordie Williamson,Oded Yacobi,Ruibin Zhang.
Symmetry is everywhere, we see it in the geometry of everyday objects, but also in differential equations or even in the laws of physics. Lie theory is the mathematical framework for understanding and using these symmetries.
One of the central problems in Lie theory is how to ‘represent’ symmetries, leading to representation theory. This theory is an important gateway for applications of Lie theory to areas like coding theory and quantum chemistry. A classical example is the representation theory of the rotation group dictating the energy level structure of the hydrogen atom.
Our new results are often obtained by establishing original connections with other fields of mathematics. For instance, we derive and study:
Seminars
Key researchers:John Cannon,Kevin Coulembier,David Easdown,Alexander Fish,,Nalini Joshi,Gus Lehrer,Andrew Mathas,James Parkinson,Bregje Pauwels,Jonathan Spreer,Anne Thomas,Stephan Tillmann,Geordie Williamson,Oded Yacobi,Ruibin Zhang.
Group theory is central to modern mathematics and its applications. The relationships between solutions to equations, positions of an object in motion, atoms in a crystal or letters in a genetic code are all described by a group, and group theory is essential to encryption.
Many other fundamental abstract structures in algebra, such as rings, fields and vector spaces, are generalisations of groups.
Research on group theory and related concepts at the University of Sydney includes both computational algebra and the investigation of theoretical questions for their own sake. Our work connects closely to algebraic geometry, representation theory, integrable systems, ergodic theory, geometric group theory and low-dimensional topology.
Seminars
Software
is a large, well-supported software package designed for computations in algebra, number theory, algebraic geometry, and algebraic combinatorics.
Key researchers:John Cannon,Kevin Coulembier,Zsuzsanna Dancso,,Gus Lehrer,Andrew Mathas,Alexander Molev,James Parkinson,Bregje Pauwels,Anne Thomas,Kane Douglas Townsend,Geoff Vasil,Geordie Williamson,Oded Yacobi,Ruibin Zhang.
Representation theory is a cornerstone of modern mathematics, as it allows us to describe symmetries of objects that are not manifestly geometric. It plays a crucial role in number theoryand has many important applications within the mathematics and the sciences, including in particle physics, cryptography, and quantum computation.
The University of Sydney employs a world-class group of researchers in representation theory, working in various fundamental areas. These include:
Representations of the symmetric group and related algebras
In particular, our researchers focus on representations of the symmetric group over fields of positive characteristic, which are not well-understood.
Geometric representation theory
This theory applies algo-geometric tools to Lie theory, the mathematical framework of symmetries.
Categorical theory and representation theory
These areas study categories of representations from both the Tannakian perspective (reconstructing the group from its representations), and higher representation theory (studying symmetries of the categories themselves).
Invariant theory
One of the oldest branches of representation theory, it studies the effect of a group acting on polynomial functions on a space or variety.
Quantum groups
Originating in statistical mechanics, quantum groups are deformations of universal enveloping algebras of lie algebras, or closely related algebras.
Key researchers:Eduardo Altmann,Dzmitry Badziahin,Chris Bertram,Nathan Duignan,Holger Dullin,Alexander Fish,Georg Gottwald,Nalini Joshi,Robert Marangell,James Parkinson,Milena Radnovic,Anne Thomas,Stephan Tillmann,Martin Wechselberger,Geordie Williamson.
Dynamics is the study of the limiting behaviour of an observable variable, such as temperature or pressure, along with time evolution. Dynamical systems are manifolds or other topological spaces endowed with a group action.
One of the first examples of a dynamical system was introduced by Newton, who studied the trajectories of planets in the solar system. While trying to prove the stability of the solar system in 1890, Poincaré initiated qualitative methods to study dynamical systems preserving volume. These insights became a cornerstone of ergodic theory.
At the University of Sydney, research on dynamical systems and ergodic theory combines the study of theoretical questions and the application of dynamical methods. Theoretical areas of dynamics includebilliards, chaos, ergodic theory and integrable systems.
Our researchers apply dynamical methods in biology, climate, complex networks,machine learning,physiology, as well as many other fields. Using dynamics, we also provide insights into other areas of pure mathematics such as additive combinatorics, algebra, geometric topology, geometric group theory and number theory.
Seminars
Key researchers:Emma Carberry,Harini Desiraju,Holger Dullin,Nalini Joshi,Robert Marangell,Jae Min Lee,Milena Radnovic,Pieter Roffelsen.
Integrable systems form the core of classical mechanics, modern mathematical physics and special function theory, arising in applications that are widespread and growing rapidly. Examples include plasma physics, elementary particles, superconductivity, and non-linear optics.
One area of intense recent activity in which integrable systems arise is the study of energy levels of heavy particles in atomic physics, which is closely connected to the spectral properties of random matrices.
This serendipitous connection has given rise to some of the most active and fruitful developments of mathematics in recent times.
Applications of random matrix theory are numerous and far-reaching: from particle physics to the distribution of airline boarding times andthe zeros of the Riemann-zeta function along the critical line.
Our research interests lie in:
Key researchers:Emma Carberry,,Nalini Joshi,Gus Lehrer,Laurentiu Paunescu,Milena Radnovic,Anne Thomas,Geordie Williamson,Oded Yacobi,Zhou Zhang.
Polynomial equations such as x3-x=y2have been at the centre of mathematics for millennia. Algebraic geometry measures the shapes formed by the solutions to systems of polynomial equations, in varying number systems.
Any high school student learns how to find the solutions to some types of polynomial equations, and sometimes how to view them geometrically. For example, the solutions to x2+y2=z2in the real numbers form a cone. But what about the solutions in the complex numbers? Or a finite field?
One of the crucial ideas in algebraic geometry is that it is very useful to consider the solutions in varying number systems simultaneously. This flexibility of viewpoint has made algebraic geometry one of the most important theoretical constructions in modern mathematics, bringing together complex analysis and number theory under one roof.
Researchers at the University of Sydney study algebraic geometry both for its intrinsic interest and for its applications to areas such as integrable systems, representation theory, and singularity theory.
Key researchers:Zsuzsanna Dancso,Laurentiu Paunescu,Jonathan Spreer,Anne Thomas,Geordie Williamson,Stephan Tillmann.
Manifolds are higher-dimensional generalisations of shapes such as a piece of string or the surface of a sphere. A small part of a manifold looks like the familiar Euclidean space, but the overall shape may be very different.
Fundamental research questions in this area are often not only interesting in their own right, but their resolution for special classes of manifolds is important for applications in mathematics and the sciences.
For example, methods to distinguish between two given manifolds can help scientists to detect mutations in DNA, or even determine the shape of our universe.
Determining natural geometric structures on a given manifold, and being able to make exact measurements, has applications to optimisation problems and magnetic resonance imaging.
Research problems of the group at the University of Sydney are often motivated by classification problems of manifolds and their distinguishing algebraic, combinatorial or geometric properties, and are related to geometric group theory, topological quantum field theories, gauge theory and algebraic geometry.
Seminars
Key researchers:Kevin Coulembier,Zsuzsanna Dancso,Gus Lehrer,Andrew Mathas,Alexander Molev,James Parkinson,Anne Thomas,Geordie Williamson,Oded Yacobi,Ruibin Zhang.
Symmetry is everywhere, we see it in the geometry of everyday objects, but also in differential equations or even in the laws of physics. Lie theory is the mathematical framework for understanding and using these symmetries.
One of the central problems in Lie theory is how to ‘represent’ symmetries, leading to representation theory. This theory is an important gateway for applications of Lie theory to areas like coding theory and quantum chemistry. A classical example is the representation theory of the rotation group dictating the energy level structure of the hydrogen atom.
Our new results are often obtained by establishing original connections with other fields of mathematics. For instance, we derive and study:
Seminars
Key researchers:Eduardo Altmann,Clio Cresswell,Georg Gottwald,Nalini Joshi,Rachel Wang.
Complex systems are abound in the natural sciences, engineering and social sciences. They involve dynamics that occurs on vast temporal and spatial scales; from Earth’s global climate, the human brain, to software systems and cities.
Network theory provides a powerful language to eke out the essential information needed to understand and control complex systems.
Given their sheer complexity, a detailed description of complex systems may not be possible, even with advanced computational tools. Mathematicians address this challenge by identifying relevant macroscopic variables and describing their dynamics or mutual relationships. Both are highly non-trivial tasks that require sophisticated mathematics.
In our research, we develop novel methodologies to improve our understanding of complex systems, with applications to gene and brain networks, language evolution, natural language processing, information spreading in social media, coupled oscillators and climate dynamics.
The task of unravelling qualitative statistical behaviour of complex systems and functional structure in networks requires us to borrow theory from dynamical systems theory, graph theory, statistical mechanics, inverse modelling and Bayesian statistics, to name but a few.
Key researchers:Eduardo Altmann,Nathan Brownlowe,Emma Carberry,Kevin Coulembier,Nathan Duignan,Holger Dullin,Ben Goldys,Daniel Hauer,Nalini Joshi,Gus Lehrer,Robbie Marangell,Alexander Molev,Milena Radnovic,Oded Yacobi,Ruibin Zhang.
Mathematical models with a physical origin are an infinite sourceof problems and inspiration for mathematicians. At the University of Sydney, our activities range from rigorous foundational work in pure mathematicsto the detailed analysis of particular models with the objective to extractnew insights in applied mathematics.
A vast set of methods from all areas in mathematics is applied to models, describing anything from the Universe to the quantum world. Structures that are first found in mathematical physics often trigger research in algebra and analysis.
Their interplay is especially evident in areas such as Hamiltonian dynamics and Lie theory. Mathematical physics has also spawned whole new fields such as integrable systems.
Mathematics and statistics not only play a central role in specialised sciences and technology sectors, but penetrate the social and behavioural sciences, as well as government, industry and management. Our researchers actively work to building mathematical and statistical knowledge in the community.
Key researchers:Emma Carberry,Daniel Daners,David Easdown,Sean Gasiorek,Daniel Hauer,Shelton Peiris,Garth Tarr,Di Warren.
Mathematical Education is a dynamic field. The teaching and learning of mathematics is complex, and the educational environment is ever changing. Maths-Ed encompasses mathematics, statistics and the relatively new field of data science.
Fundamental research questions concern how to identify and navigate the barriers to acquiring understanding in a certain area of mathematics. For example, how do students develop the concept of proof? Or how do students learn to code?
Research problems of the group at the University of Sydney span threshold concepts, statistical literacy, and best practises in teaching and learning, including mentoring new teachers and intensive and HyFlex modes of delivery.
Key leads:DmitryBadzhiahin,Emma Carberry,Clio Cresswell,Zsuzsanna Dancso,DavidEasdowne,Stephan Tillmann,Di Warren.
Staff members regularly communicate traditional and cutting-edge research to broad audiences in Sydney and NSW. These include events in partnership with theand the.
Events include the Mathematical Enrichment Day, the International Day of Mathematics on 14 March andin August.
Staff have also volunteered for the, a sub-program of STEM Professionals in Schools, supported by the Australian Government Department of Education and CSIRO.
