Research_

Dynamics and symmetry

Unveil complexity and beauty

Discover and explain nature's laws at every scale

Nature is inherently nonlinear and complex, yet it also reveals symmetry at every scale. Much of its beauty is reflected in the border between chaos and order. Here unexpected universal structures can lead to a deep understanding of nature.

The diverse research areas of the School's Dynamics and Symmetry cluster represent different ways to study and unveil the complexity and beauty of the laws of nature. These research fields complement and inform each other. Together they strive for a complete understanding of nature and social systems.

*Header image: Martin Gossow (Science Dalyell Showcase project): Limit set of a Schottky Group*

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: 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, 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: 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.

- Fluid dynamics