Research

The Topology Research Group at Melbourne has researchers working in a broad range of topological disciplines including: the interplay of geometry and the topology of manifolds in dimensions three and four, surgery theory, algebraic topology and homotopy theory.

Members:

A/Prof Diarmuid Crowley works in the differential topology of high dimensional manifolds and surgery theory.  His specific interests span the topological residues of geometric structures such as G_2-structures and contact structures, 7-manifolds, the foundations of surgery, spaces of diffeomorphism and mapping class groups and exotic spheres and the Gromoll filtration.

Dr Thorsten Hertl works in differential geometry and algebraic topology, with a focus on spaces of geometric structures and their moduli; e.g. spaces positive scalar curvature metrics or spaces of Riemannian metrics with exceptional holonomy. In this work he uses tools from combinatorial topology, stable and unstable homotopy theory, Riemannian geometry, and Index theory. He is currently investigating approaches to constructing non-trivial elements in higher homotopy groups of G_2 moduli spaces, as well as invariants for detecting such elements.

Dr Scott Mullane works in the area of moduli spaces, most frequently moduli spaces related to Riemann surfaces including the moduli space of curves, strata of differentials, and Hurwitz spaces. He is interested in how the different perspectives from topology, geometry, dynamics, and algebraic geometry relate and inform each other on these spaces.

Prof Paul Norbury works in algebraic geometry and topology with particular focus on the moduli space of curves.  His recent work is on cohomological field theories which organises algebraic topological invariants of the compactification of the moduli space of curves.  He applies the technique of topological recursion which underlies many geometric problems in mathematical physics.  He also has interest in gauge theory and the moduli space of super Riemann surfaces.

Dr Arunima Ray works on low-dimensional topology, in particular 4-dimensional manifolds and knot concordance. She is especially interested in the disparity between smooth and topological 4-manifolds, and techniques inspired by high-dimensional topology.

A/Prof Marcy Robertson works in algebraic topology and is particularly interested in using higher categories and the theory of operads to recast classical topological objects in a homotopical setting. Her recent work has focused on using higher operads to model the moduli space of genus g curves and Teichmüller space. She is also interested in using props and properads to study cohomology operations and deformation theory.

Dr TriThang Tran works in algebraic topology and is interested in homological stability, configuration spaces and braid groups. He is also engaged in research in mathematics education, teaching and learning.

Emeritus Members:

A/Prof Craig Hodgson works in hyperbolic geometry, low dimensional topology and differential geometry. His research interests include computation of geometric structures using triangulations, and the study of geometric and arithmetic invariants of hyperbolic 3-manifolds.  He has also developed a powerful deformation theory of hyperbolic structures using harmonic differential forms to represent cohomology classes.  Recently he has been investigating relationships between new invariants of 3-manifolds arising from physics, such as the 3D-index, and classical topology and geometry, including surfaces and hyperbolic structures.

Prof J. Hyam Rubinstein works in low dimensional topology and is especially interested in three and four-dimensional manifolds. He is also interested in studying shortest networks with applications to mining, machine learning and financial mathematics.

Researchers in Related Areas:

A/Prof Nora Ganter works on Moonshine, categorification and is interested in the applications of elliptic cohomology to representation theory.

Prof Christian Haesemeyer works in the area of algebraic K-theory and motivic homotopy theory. He often uses techniques from algebraic topology and homotopy theory and has an ongoing interest in surgery theory as it relates to quadratic forms.

Dr Yaping Yang works in the area of Lie algebras and quantum groups. She has worked on the related geometry and topology of quiver varieties, and affine Grassmannians arising from representation theory.

Dr Gufang Zhao works on oriented cohomology theories and applications in representation theory and enumerative geometry. He is also interested in structures of moduli of sheaves on Calabi-Yau 3-folds.