# Geometry and Topology

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.

## Seminar series

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
Prof Christian HAESEMEYER (Professor)
Research interests: K-theory, Algebraic cycles, Motives
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: Gauge Theory, Mathematical Physics, Algebraic geometry, Moduli spaces
Dr Sam POVALL (Outreach Fellow)
Research interests: Geometric group theory, Singularity Theory, Complex Hyperbolic Geometry
Dr Lawrence REEVES (Senior Lecturer)
Research interests: Geometric group theory
Dr Marcy ROBERTSON (Senior Lecturer)
Research interests: Higher category theory, Operad theory, Homotopy Theory
Dr Volker SCHLUE (Lecturer)
Research interests: Partial Differential Equations, General relativity, Geometric analysis
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 (Senior Lecturer)
Research interests: Algebraic groups, Representation theory
Dr Yaping YANG (Lecturer)
Research interests: Geometric representation theory, Quantum groups
Dr Gufang ZHAO (Lecturer)
Research interests: Geometry & Topology, Algebra

## Research Fellows

Dr Gufang ZHAO (ARC DECRA)
Research interests: Algebra, geometry, topology, Representation theory, Topology, Algebraic geometry

## 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)

CHAIMANOWONG Wee – ‘Nekrasor Partition Function and Frobrnius Manifolds.
COLMAN Owen
ELLIOT Patrick
HAN Songqi
LIU Jayson
MACLEOD Nicholas
NAGY Csaba
RAHMAN Anas
STRUMILA Michelle

ANAGNOSTOU Lukas
BROWN Jason
CLARK Alexander
DOVE Thomas
FIELD Tyson
HOGAN Tamara