# Geometry, Topology and Analysis

## Research Interests of the GTA Group

This group has wide interests in the broad areas of Analysis, Geometry and Topology, which lie at the heart of much of Mathematics.

- Dr Carberry works on geometric aspects of integrable systems, in particular the role that integrable systems play in allowing one to apply techniques from algebraic geometry to differential-geometric problems.
- Dr Cartwright works on harmonic analysis for groups acting on trees and buildings, and on the theory of random walks on graphs and groups.
- Dr Fish works in ergodic theory, with particular emphasis on ergodic theoretical problems related to additive combinatorics of discrete amenable groups. Dr Fish is also interested in applications of harmonic analysis and representation theory to problems in wireless communication.
- Dr Hillman is working on the algebraic topology of low-dimensional manifolds, and in particular is interested in the influence of Poincaré duality on the fundamental group.
- Dr Kuo and A/Prof Paunescu are working on singularity theory and algebraic geometry over the classical fields.
- Dr Parkinson works in harmonic analysis and representation theory, with a particular emphasis on algebraic and geometric structures related to p-adic Lie groups, such as affine buildings and affine Hecke algebras.
- Dr Radnović works in geometry and integrable systems. Her research interests include geometry of families of quadrics, theory of algebraic curves, their Jacobians and theta-functions, mathematical billiards, topological classification of classical integrable systems, Painlevé equations.
- Dr Thomas works in geometric group theory. She uses a variety of methods to investigate polyhedral complexes, including buildings, lattices in automorphism groups of such complexes, and related topics such as Coxeter groups and group algebras.
- Dr Tillmann explores the geometry and topology of manifolds using a broad range of techniques including different flavours of geometry (hyperbolic, projective, algebraic and tropical) as well as group theory and triangulations.
- Dr Zhang works on nonlinear partial differential equations coming from differential geometry consideration (for example, Ricci flow and the complex version of it).

## Research Areas

Algebraic invariants of 4-manifolds, knot theory, homological group theory, harmonic analysis, buildings, real and complex singularities, stratifications and subanalytic sets, Hodge theory, topology of algebraic varieties, surface theory, spectral curves, integrable systems, geometric evolution equations, pluripotential theory, geometric group theory.

## Contact Person

A/Prof L. Paunescu (laurentiu.paunescu@sydney.edu.au)