Dr Jonathan Spreer

The University of Sydney

Selected research projects

2019 - 2021

Hyam Rubinstein, Jonathan Spreer, Stephan Tillmann, Trisections, triangulations and the complexity of manifolds, Australian Research Council (ARC) Discovery Project 2019, DP190102259.



Topology is the mathematical study of the shape of spaces such as surfaces and their higher dimensional analogues. Geometry endows these spaces, also called manifolds, with additional properties such as distance, angle and curvature. Manifolds can be studied by decomposing them into simple pieces such as triangles. These triangulations are a very effective tool. Typically the fewer pieces they have, the better they describe their underlying manifold. We develop a new approach based on such small triangulations that aims at practical algorithms for manifolds in dimensions 3 and 4. Concrete aims include connectivity results for triangulations, and algorithms to recognise fundamental topological structures such as trisections and bundles.

2015 - 2020

Benjamin A. Burton, Jonathan Spreer, Tractable topological computing: Escaping the hardness trap, Australian Research Council (ARC) Discovery Project 2015, DP150104108.



Computational topology is a young and energetic field that uses computers to solve complex geometric problems driven by pure mathematics, and with diverse applications in biology, signal processing and data mining. A major barrier is that many of these problems are thought to be fundamentally and intractably hard. This project will defy such barriers for typical real-world inputs by fusing geometric techniques with technologies from the field of parameterised complexity, creating powerful, practical solutions for these problems. It will shed much-needed light on the vast and puzzling gap between theory and practice, and give researchers fast new software tools for large-scale experimentation and cutting-edge computer proofs.

2014 - 2016

Benjamin A. Burton, Basudeb Datta, Jonathan Spreer, Nitin Singh, Building triangulations for fast topological computing, DIICCSRTE, Australia-India Strategic Research Fund (AISRF), Round 7, AISRF06660.

Elliptic curve


Computational topology is a young and fast-growing area of ICT, with roots in geometry and applications in biology, physics, computer vision and cosmology, in which real computations are often prohibitively expensive. We will overcome this by building "tight triangulations", highly efficient forms of input with which we can solve substantial problems cheaply using new and innovative heuristics. Outcomes will include practical software, with significant benefits spanning both ICT and mathematics.

Selected conferences and workshops

27-31 July 2020

Computational & Algorithmic Topology, Sydney (CATS 2020)

This workshop at The University of Sydney will bring together experts and emerging researchers from Australia, the USA and Europe to report on recent results and explore future directions in computational and algorithmic topology and related areas. There will be a focus on problems in computational geometric topology and computational algebraic topology. This workshop aims to stimulate interaction between researchers in order to bring about new collaborations on difficult problems that cannot be tackled from one viewpoint alone.


29-31 October 2018

Einstein Workshop Geometric and Topological Combinatorics

This will be a three day workshop on Geometric and Topological Combinatorics, with lectures in the morning and time for discussions and networking in the afternoons. It is primarily funded by the Einstein Foundation Berlin, as part of Francisco Santos' Einstein Visiting Fellowship within the Discrete Geometry Group at FU Berlin for 2016-2019. Additional funding is provided by the project Discretization in Geometry and Dynamics (SFB Transregio 109) and the Berlin Mathematical School.


13-16 March 2018

Einstein Workshop Discrete Geometry and Topology

This will be a four day workshop with lectures, presentations and informal discussions on discrete geometry and topology, and related topics (combinatorial topology, computational geometry, polytope theory). It is primarily funded by the Einstein Foundation Berlin, as part of Francisco Santos' Einstein Visiting Fellowship at FU Berlin for 2016-2019. Additional funding is provided by the project Discretization in Geometry and Dynamics (SFB Transregio 109) and the Berlin Mathematical School.


Selected talks (with slides)

Oct 2019 Talk at the Pure Maths Seminar, The University of Queensland, Brisbane, Australia.
Title: Linking topology and combinatorics: Width-type parameters of 3-manifold
slides ]
May 2019 Talk at Sydney Algorithms Group Seminar, The University of Sydney, Australia.
Title: Discrete Algorithms for Geometric Topology
slides ]
Dec 2015 Talk at the 39th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, Brisbane, Australia.
Title: Separation index of graphs and stacked 2-sphere
slides ]
Oct 2015 Talks at Institute for Science and Technology, Klosterneuburg, Austria; Technische Universität Berlin, Germany; Indian Institute of Science, Bangalore, India; and Indian Statistical Institute, Calcutta, India.
Title: Algorithms and complexity for Turaev-Viro invariants
slides ]
Oct 2015 Presentation at Mathematical Research Institute Oberwolfach, see extended abstract above.
Title: Random collapsibility and 3-sphere recognition
slides (from the 2015 AustMS meeting in Adelaide) ]
Jun 2015 Talk at Freie Universität Berlin, Germany.
Title: Parameterised complexity theory for topological problems
slides (from the 2013 AustMS meeting in Sydney) ]
Feb 2015 Talk at University of Ulm, Germany.
Title: Geometrie und Topologie im Computeralgebrasystem GAP
slides ]
May 2014 Talk at the National Institute of Informatics Shonan Meeting on Knot theory, Algorithms, complexity and computation, Tokyo, Japan.
Title: Tightness for triangulations
slides (from the 2014 AustMS meeting in Melbourne) ]
Apr 2014 Talk at the Computational and Algorithmic Topology workshop (CATS), Sydney, Australia.
Title: Bounds for the genus of a normal surface
slides ]

Dr Jonathan Spreer   |   The University of Sydney   |   School of Mathematics and Statistics   |   Email: jonathan.spreer@sydney.edu.au