# Analysis, geometry and topology research

The research areas of analysis, geometry and topology have strengths in differential geometry, functional analysis, harmonic analysis and topology.

## Research topics

### Applications of topology to analysis

In this research topic, we apply general topology to problems in analysis and functional analysis. These two areas played fundamental roles in the birth of general topology.

At the interface of these two disciplines are several interesting concepts and results.

### Conformal geometry, submanifolds, and natural partial differential equations

Natural partial differential equations (PDE) are equations whose coefficients are geometrically determined by the underlying structure on which they are defined.

They provide the primary tool for studying global structures via analysis and a vast array of physical and natural phenomena from particle equations to biological systems.

### Degenerate operators

The heat flow is well understood when the conductivity is positive (but not necessarily smooth), and when the conductivity is smooth but vanishes at some parts of the material.

We aim to understand the global behaviour of heat flow if the conductivity is both non-smooth and may vanish on some parts.

### Generalised inverse limits

Mathematicians have studied inverse limits extensively for many years. However, the introduction of generalised inverse limits (GILS) is relatively recent. They involve subsets of infinite dimensional spaces, working backwards through (bonding) functions to find their coordinates.

Our research studies the relationship between the properties of inverse limits and the properties of their bonding functions.

### Pluripotential theory

We study potential theory on general spaces in several complex variables.

Our research replaces the complex plane by more general spaces. In particular, algebraic varieties, which are geometric sets given by solutions to polynomial equations.

### Tight frames and their symmetries

If one of two cartesian coordinates for a battleship is lost, then its position can no longer be determined. However, it is possible to give three coordinates so that its position can still be determined if one is lost (this is more efficient than repeating both coordinates twice). This is an example of what is called a tight frame.

Tight frames offer natural generalisations of orthonormal bases which reflect additional symmetries of the space.

## Our researchers

#### Dr Pedram Hekmati

• Differential geometry
• Lie theory
• Mathematical physics

#### Professor Rod Gover

• Differential geometry
• Analysis on manifolds
• Partial differential equations and mathematical physics
• Conformal and related structures

#### Associate Professor Sina Greenwood

• Continua theory
• Discrete dynamical systems

#### Dr Sione Ma'u

• Pluripotential theory and its applications
• Functions of several complex variables

#### Associate Professor Warren Moors

• Functional analysis
• Applications of topology to analysis

#### Professor Tom ter Elst

• Harmonic analysis
• Operator theory
• Geometric analysis
• Sub-elliptic and degenerate operators
• Partial differential equations

#### Dr Shayne Waldron

• Finite tight frames with symmetries
• Approximation theory
• Polynomial interpolation
• Numerical methods

### Notable visitors

The Department of Mathematics hosts many international visitors, some of our notable visitors in the field of analysis, geometry and topology include:

• Professor Wolfgang Arendt (University of Ulm)
• Professor Len Bos (University of Verona)
• Professor Andreas Cap (University of Vienna)
• Professor Paul Gartside (University of Pittsburgh)
• Professor Judy Kennedy (Lamar University)
• Professor Zengjian Lou (Shantou University)