Organisers:
Jessica Purcell (Monash University)
Marc Lackenby (University of Oxford)
Jonathan Spreer (University of Sydney)
Adele Jackson (University of Sydney)

Program Description:
The mathematical study of surfaces and manifolds in dimensions three and four have a rich history of more than a century, dating back at least to Poincaré. One tool that has become increasingly important in their study is that of triangulations and similar decompositions. A manifold is described as a gluing of simpler pieces, such as simplices. The combinatorics and geometry of the decomposition can be used to prove important results about the manifold.

In the 1960s and 1970s Haken used triangulations to consider how surfaces lie in 3-manifolds; applications of his pioneering work continue to produce new insights. In the 1970s and 1980s, following ideas of Thurston, Neumann and Zagier used geometric triangulations to determine hyperbolic-geometric properties of topological Dehn fillings. But the existence in general of geometric triangulations is still unproven. In recent decades, triangulations have become one of the most useful ways to represent 3- and 4-manifolds in computing. This has lead to a more algorithmic study of low-dimensional topology, driven by the combinatorics of triangulations.
As a result, a wealth of new questions and problems have emerged. Examples include open questions on the complexity of 3-manifolds, on effective algorithms for 4-dimensional manifolds, on the search for triangulations that are either good or pathologically bad, etc. There are also open questions about the geometric and topological properties of triangulations, and what geometry implies about the topology, and vice-versa.

The purpose of this workshop is to take a week to bring everyone together, along with Australia’s next generation of researchers in the field, to investigate and tackle these challenges.

Program Structure:
TBD

Registration:

• Deadline: TBD
• Registration is by invitation only. If you are interested to participate in this program, please contact one of the organisers with your CV and research background.
• Arrival date: 1 June 2025

ASSOCIATED EVENTS
MATRIX Wine and Cheese Afternoon 3 June 2025
On the first Tuesday of each program, MATRIX provides a pre-dinner wine and cheese afternoon. Produce is locally-sourced to showcase delicacies from the region.