Funding

Major research grants (current)

2022 - 2025

Jonathan Spreer, Francisco Santos, Triangulations: linking geometry and topology with combinatorics, Australian Research Council (ARC) Discovery Project 2019, DP220102588. A$433.000

Abstract: Triangulations are the method of choice to represent geometric objects given by a finite sample of points. Prominent examples include the pictures produced by the finite element method, polytopes in optimisation, or surfaces in computer graphics.

Knowledge about the triangulations of an object and how they relate to each other is essential for these applications. Seemingly canonical and straightforward methods perform well - or not at all, depending on intricate and highly involved mathematical properties.

In this project we combine geometric and topological viewpoints to tackle high-profile questions about triangulations. This will unlock the full potential of combinatorial methods and practical algorithms in applications.

2019 - 2023

Stephan Tillmann, Hyam Rubinstein, Jonathan Spreer, Trisections, triangulations and the complexity of manifolds, Australian Research Council (ARC) Discovery Project 2019, DP190102259. A$395.000

Abstract: 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.

Major research grants (past)

2015 - 2021

Benjamin A. Burton, Jonathan Spreer, Tractable topological computing: Escaping the hardness trap, Australian Research Council (ARC) Discovery Project 2015, DP150104108. A$367.000

Abstract: 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. A$295.000

Abstract: 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.