This is the project page for the Australian Research Council funded Discovery Project

Triangulations in dimension three:

algorithms and geometric structures

ARC Discovery Grant DP1095760

2010-2012

## personnel

- Chief Investigators:
- Craig Hodgson (Melbourne)
- Hyam Rubinstein (Melbourne)
- Stephan Tillmann (Queensland 2010-2011; Sydney 2012)
- Research Fellow:
- Henry Segerman
## summary

The aim is to constructively build the optimal geometric shapes for three-dimensional spaces and study their properties by new practical algorithms. It is known, by remarkable work of Perelman, that all three-dimensional spaces can be decomposed into pieces admitting uniform geometric structures. However a constructive way of finding these structures is not known. We will attack this problem by finding suitable ways of splitting each space into tetrahedra which carry the geometry. This will enable us to construct and implement new practical algorithms to recognise different three-dimensional spaces and to identify important surfaces in them, as needed in knot theory. ## publications

- Veering triangulations admit strict angle structures

Geometry & Topology 15 (2011) 2073-2089.

Authors: Craig Hodgson, Hyam Rubinstein, Henry Segerman, Stephan Tillmann

- Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds

Proceedings of the American Mathematical Society, in press.

Authors: Feng Luo, Stephan Tillmann, Tian Yang

- The Weber-Seifert dodecahedral space is non-Haken

Transactions of the American Mathematical Society 364 (2012), no. 2, 911-932.

Authors: Ben Burton, Hyam Rubinstein, Stephan Tillmann

- Finiteness of 3-manifolds associated with non-zero degree mappings

Commentarii Mathematici Helvetici, in press.

Authors: Michel Boileau, Hyam Rubinstein, Shicheng Wang

- Pseudo-developing maps for ideal triangulations I: Essential edges and generalised hyperbolic gluing equations

Contemporary Mathematics 560 (2011), 85-102.

Authors: Henry Segerman, Stephan Tillmann

- Projective Deformations of Hyperbolic Coxeter 3-Orbifolds

Geometriae Dedicata, DOI:10.1007/s10711-011-9650-8.

Authors: Suhyoung Choi, Craig Hodgson, Gye-Seon Lee

## preprints

- On Convex Projective Manifolds and Cusps, submitted for publication.

Authors: Daryl Cooper, Darren Long, Stephan Tillmann

- Triangulations of hyperbolic 3-manifolds admitting strict angle structures, submitted for publication.

Authors: Craig Hodgson, Hyam Rubinstein, Henry Segerman

## work in progress

- Triangulations of 3-manifolds with essential edge loops

Authors: Craig Hodgson, Hyam Rubinstein, Henry Segerman, Stephan Tillmann- On the number of hyperbolic 3-manifolds with a given volume

Authors: Craig Hodgson, Hidetoshi Masai- On minimal ideal triangulations

Authors: William Jaco, Kei Nakamura, Hyam Rubinstein, Stephan Tillmann

- Constructing hyperbolic structures on 3-manifolds using twisted tetrahedra

Authors: James Dowty, Craig Hodgson and Peter Milley

## code contributions

- Boundary curves of spun-normal surfaces; implemented in Regina (version to be released May 2012)
- Transversely oriented normal surfaces; implemented in Regina (version to be released May 2012)
- Transversely oriented spun-normal surfaces; implemented in Regina (version to be released May 2012)
## conferences

- 2010
- Algorithms, Algebra and Analysis in Four Dimensions, Brisbane, Australia (20-21 July)
- 2011
- Geometry & Topology Down Under, Melbourne, Australia (11-22 July)
## visitors

- 2010
- Joel Hass (UC Davis) Melbourne: 20.1.-30.6.
- Abigail Thompson (UC Davis) Melbourne: 20.1. - 30.6.
- 2011
- Peter Scott (Michigan) Melbourne: 9.2.-16.3.
- Hidetoshi Masai (Tokyo Institute of Technology) Melbourne: 9.7.-5.10.
- Ian Agol (UC Berkeley) Queensland: 24.7.-27.7.
- Joseph Maher (CUNY Staten Island) Queensland: 27.7.-8.8.
- Bus Jaco (Oklahoma State) Queensland: 8.8.-16.8.
- 2012
- Joseph Maher (CUNY Staten Island) Sydney: June/July
## students

- Shanil Ramanayake (PhD student)

Supervisors: Craig Hodgson, Hyam Rubinstein- Ahmad Issa (MSc student)

Supervisor: Craig Hodgson## related links

- Nathan Dunfield's CompuTop.org Software Archive
- Jeff Weeks' Topology and Geometry Software
## related news

- The Clay Institute awarded the first Millennium Prize to Grigoriy Perelman for the resolution of the Poincaré Conjecture.

http://www.maths.usyd.edu.au/u/tillmann/DP1095760/

18 February 2012