Research_

# Geometry and topology

New insights about the shapes and structures of spaces
The nexus of diverse research areas such as algebraic geometry, differential geometry, and complex systems.

The study of geometry has its roots in ancient times. Topology, the mathematical study of the shape of spaces, has its origins in the nineteenth century.

Today, geometry and topology are intertwined thriving fields of research that offer new insights into many different branches of mathematics and theoretical physics. They are a nexus of diverse research areas such as algebraic geometry, differential geometry, and complex systems. Here, these areas meet and develop a synergy, they are influenced by geometry and topology, and enrich geometry and topology.

Such interaction between work in geometry and topology and neighbouring areas is also true of the group at the University of Sydney.

Header image: Inside view of the Weber-Seifert Space, produced with Curved Spaces by Jeff Weeks

### Research areas

#### Algebraic geometry

Polynomial equations such as x3-x=y2 have 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=z2 in 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.

#### Manifolds and cell complexes

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

#### Lie theory

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

SMRI Algebra and Geometry Online

Algebra Seminar

Informal Friday Seminar

#### Networks and complex systems

Key researchers: Eduardo AltmannClio CresswellGeorg GottwaldNalini JoshiRachel 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.

• Differential geometry
• Algebraic topology
• Mathematical physics