Research_

Algebra and structures

Uncover structure where chaos seems to reign

Gain new insights and enable discoveries by detecting and analysing structures in natural phenomena, data sets and abstract theories

A key role of modern mathematics is detecting and analysing structures in a broad range of domains, from particle physic to cosmology, data sets to neural networks. New insights and predictions are often only made possible from a deep understanding of an abstract theoretical framework, and uncovering structure where chaos seemed to reign.

The interaction of algebra—one of the oldest structure theories—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.

Header image: Anne Thomas: *“Folded Galleries in an affine building”*

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:

- geometric realisations of (categories of) representations,
- combinatorial tools for describing the structure of representations,
- higher categorical representation theory, to deal with structures that resist satisfactory classical representations,
- dualities between representation theory for different types of algebraic structures.

**Seminars**

Key researchers: John Cannon, Kevin Coulembier, David Easdown, Alexander Fish, Anthony Henderson, 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**

Magma 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.

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, Anthony Henderson, 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 theory and 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 6^{th} century 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 20^{th} century, 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 and the zeros of the Riemann-zeta function along the critical line.

Our research interests lie in:

- classical mechanics
- Hamiltonian dynamics
- Painlevé equations
- discrete integrable systems
- geometry and asymptotics of integrable systems
- topological methods in integrable systems
- partial differential equations
- mathematical billiards
- harmonic maps and conformal surface theory.

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 include billiards, 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, Nicole Sutherland, Don Taylor, Bill Unger, John Voight, Geordie Williamson

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 Magma, 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.