Research Group Details

Geometry & Topology

About

To understand a basic difference between topology and geometry, imagine a circle. Now ask yourself: Is it a perfect circle? If so, what radius is it? If you cannot immediately answer these questions, your concept of a circle is topological, to do with form rather than precise rigid/geometric shape. A topological circle can manifest geometrically in many different ways ... as an almost perfect circle (as drawn by someone with a steady hand), as ellipses of different shape and size, as pieces of string with ends tied together ... But another thing is very clear: a "circle" is different from a line or an interval. In a similar way, the 2-dimensional surfaces of a ball or donut are intrinsically different, no matter how you try to stretch or distort. Each in its own way nonetheless has very nice possible geometric shapes: a sphere may manifest as a soap bubble wobbling in the air, but also tries to form a perfectly uniformly curved surface: a "perfect" sphere. On the other hand, if you slice an actual donut into two pieces with a knife, several possibilities arise for the general "shape" of the two pieces ... they may both be rings, may both be like bent balls, one bit may be a ball and the other like the original donut ... all depending on how the cut was made. There are only finitely many topological possibilities, but infinitely many geometric ones. The research group at Melbourne studies the interplay of geometry and topology, as well as some applications to processes in industry, and models of biological systems. For example, DNA can form closed loops, and there are serious difficulties understanding or modelling enzyme actions which permit the unlinking of two copies after replication. Recent work in mathematical physics of elementary particles and statistical mechanics has provided some ways to try to understand this phenomena. Related mathematics also occurs when considering the "perfect" geometric forms of possible 3-dimensional topological spaces. The most important -- hyperbolic geometry -- is intimately connected with geometric transformations which occur in the theory of relativity. When equations are used to describe things, such as the relationships of different lengths of components of a mechanical robot, the geometry and topology of possible shapes is again important to understand and describe: Fixing a length is akin to slicing a donut as described above. Problems related to combinatorial optimization often appear, which commonly are of great interest in industry. Being able to describe, display and compute such objects is also an active research area in this group.

Academic Staff

A/Prof Diarmuid CROWLEY (Associate Professor)
Research interests: Differential topology, Algebraic topology, Surgery classification of manifolds
Dr Nora GANTER (Senior Lecturer)
Research interests: Categorification, Elliptic cohomology, Homotopical representation theory, Monstrous moonshine
Dr Jesse GELL-REDMAN (Lecturer)
Research interests: Microlocal analysis, Partial Differential Equations, Differential geometry
Dr David GEPNER (Senior Lecturer)
Research interests: Algebraic K-theory, Algebraic geometry, Homotopy Theory, Higher category theory
Prof Christian HAESEMEYER (Professor)
Research interests: K-theory, Algebraic cycles, Motives
A/Prof Craig HODGSON (Associate Professor/Reader)
Research interests: Hyperbolic geometry, Low dimensional topology, Differential geometry
Dr Daniel MURFET (Lecturer)
Research interests: Mathematical logic, Algebraic geometry, Topological field theories
Prof Paul NORBURY (Professor)
Research interests: Mathematical Physics, Integrable systems, Algebraic geometry, Moduli spaces
Dr Lawrence REEVES (Senior Lecturer)
Research interests: Geometric group theory
Dr Marcy ROBERTSON (Lecturer)
Research interests: Homotopy Theory, Operad theory, Higher category theory
Dr TriThang TRAN (Senior Tutor)

Prof Kari VILONEN (Professor)
Research interests: Representation Theory, Algebraic geometry, Algebraic analysis
Dr Chenyan WU (Lecturer)
Research interests: Representation Theory, Algebraic geometry, Algebraic number theory
Dr Ting XUE (Lecturer)
Research interests: Representation Theory, Algebraic groups
Dr Yaping YANG (Lecturer)
Research interests: Geometric representation theory, Quantum groups

Research Fellows

Dr Xing GU (Research Fellow)
Research interests: Homotopy Theory, Algebraic topology
Dr Iva HALACHEVA (Research Fellow)
Research interests: Low dimensional topology, Knot theory, Representation Theory, Quantum algebra
Dr Kazuya KAWASETSU (Research Fellow)

Dr Arik WILBERT (Research Fellow)
Research interests: Topological Quantum Field Theory, Low dimensional topology, Categorification, Geometric & Combinatorial Representation Theory
Dr Gufang ZHAO (Research Fellow)
Research interests: Algebraic geometry, Topology, Representation Theory

Honorary Staff

Dr Iain AITCHISON
Senior Fellow (Associate)
Dr Dave COULSON
Fellow (Associate)
Emeritus Professor J. Hyam RUBINSTEIN
Professorial Fellow (Associate)
Professor Henry SEGERMAN
Fellow (Associate)
Dr Craig WESTERLAND
Fellow (Associate)

Visitors

Dr Zhen HUAN (Sun Yat-Sen University)
Dr Cheng-Chiang TSAI (Stanford University )

Postgraduate Students

CHAIMANOWONG Wee – ‘Nekrasor Partition Function and Frobrnius Manifolds.
ELLIOT Patrick
EPA Narthana – ‘Platonic - 2 Groups
HAN Songqi
ISSA Ahmad – ‘Low dimensional topology and geometry
KOYAMA Musashi
LEIGH Oliver – ‘Enumerative problems in algebraic geometry motivated from physics.
MCGAHEN Adrian
NAGY Csaba
SPONG Matthew
STRUMILA Michelle
WOOD Adam

Masters (RT) Students

AGISILAOU Adrian
ARMITAGE Ethan
CLIFT James
DADD Blake
ELLIOT Patrick
FIELD Tyson
LYONS Samuel
MUSGRAVE-EVANS Curtis
WINNING Heath