# Analysis & geometry

The Analysis & Geometry research program explores the areas of differential geometry, several complex variables, non-commutative geometry, harmonic analysis, microlocal analysis, partial differential equations, operator theory, spectral theory, convex geometry and statistical learning theory, and banach algebras.

## About

The Mathematical Analysis research groupÂ in MSI researches fundamental questionsÂ about functions and continuous processes. We study partial differential equations, harmonic analysis, microlocal analysis, stochastic analysis, mathematical physics, abstract analysis, geometric analysis, differential geometry, symplectic geometry and noncommutative geometry.

### Capabilities

Each faculty member runs their own independent research program, often involving postdocs, PhD students, masters students and honours students, and with numerous international collaborators.

- Ben Andrews: geometric partial differentialÂ equations (including mean curvature flow and Ricci flow), geometric variationalÂ problems (including isoperimetric problemsÂ and eigenvalue inequalities), and geometricÂ approaches to the analysis of differential equations.
- Xu-Jia Wang: fully nonlinear elliptic equations of Monge-Ampere type and their applications in geometry and physics.
- Shahar Mendelson: High-dimensional probability, theoretical data science and statistical learning theory.
- Andrew Hassell: microlocal analysis applied to spectral theory, quantum chaos, nonlinear dispersive equations, and rough wave equations.
- Pierre Portal: partial differential equations analysed using stochastic analysis to understand random dynamics at the microscopic level, and harmonic analysis to understand oscillatory patterns at the macroscopic level.
- Po Lam Yung: harmonic analysis, particularlyÂ function spaces (such as Sobolev spaces),Â singular integrals, oscillatory integrals, and related problems in Fourier restriction theory.
- Bryan Wang: geometric properties of moduli spaces in symplectic topology and gauge theories in low dimensions as well as invariants from these moduli spaces.
- Galina Levitina: Noncommutative geometry, spectral theory, operator theory and mathematical physics.
- Brett Parker: Symplectic geometry and Gromov-Witten and related invariants using exploded manifolds.
- James Tener: Operator algebras, functional analysis, and representation theory to studyÂ the mathematical foundations of quantum fieldÂ theory

### Reasons to workÂ with us

Work with world-leading mathematical analysts. The quality of our faculty is shown by numerous awards and honours, including three Laureate Fellowships, two Future Fellowships, three Fellows of the Australian Academy of Science, two ICM addresses, medals including the Morningside Medal, Ambrosetti Medal, and several Australian Mathematical Society Medals.

Many of our PhD graduates have gone on to stellar careers in academia, while others have had outstanding careers in industry and the private sector.

## Projects

## Members

### Convenor

### Emeritus

### Researcher

Honorary Professor

MSI Fellow

Postdoctoral Fellow

Postdoctoral Fellow

Professor

Postdoctoral Fellow

### Retired Fellow

Retired Fellow

Retired Fellow