# Geometry and analysis

Mathematicians researching geometry and analysis are undertaking projects with postgraduate opportunities.

The geometry and analysis research areas belong largely to the domains of continuous mathematics and number fields, having strong connections with calculus but also incorporating many ideas from algebra.

Research currently undertaken by mathematicians in this group includes Banach algebras, C*-algberas, elliptic curves, functional and global analysis, invariant theory and singularity theory with applications to robotics, and non-standard analysis.

## Researchers

**Professor of Mathematics**

**School of Mathematics and Statistics**

### Lisa Clark

**Director, Centre for Science and Society**

**Wellington Faculty of Science**

### Martino Lupini

**Professor of Mathematics**

**School of Mathematics and Statistics**

**Senior Lecturer in Mathematics**

**School of Mathematics and Statistics**

### Iain Raeburn

## Current projects

### Through the looking glass: sharpening the classification program through implications for operator algebras of graphs and groupoids (Marsden Fund: Prof Astrid an Huef, AProf Lisa Clark, Prof Iain Raeburn)

The classification program for operator algebras has held the attention of top experts for over two decades, and a great deal of progress has been made. With each forward step, the focus of the programs shifts and different constructions provide the crucial examples. Currently, there is immense interest in properties called quasidiagonality, finite nuclear dimension, and Cartan subalgebras.

The PIs will investigate these properties in the context of operator algebras associated to higher-rank graphs and groupoids. They are particularly intrigued by the possibility that there is a dichotomy for the algebras of higher-rank graphs: that they are either quasidiagonal or purely infinite.

### Logic and C*-algebras (Marsden Fund: Dr Martino Lupini)

Operator algebras and noncommutative dynamics play a crucial role in the mathematical formalization of quantum physical systems. The classification of C*-algebras and group actions, that is, the problem of obtaining explicit lists and complete invariants for such objects, is thus a central problem in modern mathematics. In recent years there has been a wave of applications of logic to operator algebras, which has seen problems settled that were hitherto out of reach of classical methods.

The main goal of this project is to solve major open problems in operator algebras by using tools from logic.

## Other projects/completed projects

Geodesics in diffeomorphism groups: geometry and applications (Prof Stephen Marsland, previous Marsden).

## Postgraduate opportunities

There are a variety of scholarships available for students studying in the Wellington Faculty of Science.

Students interested in postgraduate research can study towards an MSc by thesis or a PhD under the supervision of staff members in the School of Mathematics and Statistics.

Further information about PhD study, including scholarship funding, is available on the website of the Faculty of Graduate Research.

In addition, some staff members may have grant funding for PhD research on specific projects.

Prospective research students are encouraged to contact potential supervisors directly.