# Geometry and 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.

## Seminar series

Topology

Pure Mathematics

Algebraic Geometry and K-Theory

## 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: *Homotopy Theory, Algebraic K-theory, Algebraic geometry, 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: *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 Iva HALACHEVA** (Research Fellow)

Research interests: *Low dimensional topology, Knot theory, Representation theory, Quantum algebra*

**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)

## Postgraduate Students

CHAIMANOWONG Wee – ‘Nekrasor Partition Function and Frobrnius Manifolds.’

ELLIOT Patrick

HAN Songqi

MCGAHEN Adrian

NAGY Csaba

RAHMAN Anas

STRUMILA Michelle

SWADDLE Michael

WOOD Adam

## Masters (RT) Students

BROWN Jason

CLARK Alexander

DOVE Thomas

FIELD Tyson

HOGAN Tamara

MACHIDA Kai

MUSGRAVE-EVANS Curtis

SIMS Blake

TROIANI William

TSIGLOPOULOS Matthew

TURVEY Caelan

WALSH Adam

WINNING Heath