| since 2021 |
Senior Lecturer School of Mathematics and Statistics, The University of Sydney. | |
| 2019 - 2020 |
Lecturer and Early Career Development Fellow School of Mathematics and Statistics, The University of Sydney. | |
| 2017 - 2019 |
Postdoctoral research fellow funded by the Einstein Foundation. Einstein Visiting Fellowship Santos at Discrete Geometry Group, Freie Universität Berlin. | |
| 2016 |
Lecturer, funded by the Australia-India Strategic Research Fund (AISRF). Grant AISRF06660, at the School of Mathematics and Physics, University of Queensland. | |
| 2014-2015 |
Postdoctoral Research Fellow, funded by the Australia-India Strategic Research Fund (AISRF). Grant AISRF06660, at the School of Mathematics and Physics, University of Queensland. | |
| 2012-2013 | Postdoctoral Research Fellow at the School of Mathematics and Physics, University of Queensland. | |
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| 2011 |
Doctorate in natural sciences (Dr. rer. nat.) at the University of Stuttgart. Dissertation: Blowups, slicings and permutation groups in combinatorial topology (Doctoral advisor: Prof.W.Kühnel). | |
| 2008 |
Diploma in mathematics and computer science at the University of Stuttgart. Diploma thesis at the Institute of Geometry and Topology, University of Stuttgart: On the Topology of combinatorial 4-manifolds, in particular the K3-Surface (Supervisor: Prof.W.Kühnel). | |
| 2007 | Maîtrise (french one year postgraduate diploma) in mathematics at the Université Pierre et Marie Curie in Paris. | |
My research interests are motivated by the study of manifolds (i.e., surfaces and their higher-dimensional analogues). For this, I use a blend of techniques from mathematical fields such as low-dimensional topology and combinatorics, tools from computer science, such as parameterised complexity theory, as well as practical skills from algorithm design and the development of mathematical software.
If you are thinking about doing a research project or honours project in computational geometry and topology (or any related field) - or even if you just want to know more about my research - please do not hesitate to contact me.
Possible research projects (all levels):
1. Highly symmetric triangulations of manifolds (surfaces and their higher dimensional analogues): Given a surface decomposed into triangles (or, more generally, n-gons), edges and vertices, there is a natural upper bound on the number of symmetries. Decomposition attaining this upper bound are very famous objects which often occur in many other settings as well. Examples for such objects are the platonic solids.
While the platonic solids are very well-known and studied, this is not so much true for other symmetric decompositions of surfaces and higher dimensional manifolds. This is despite the fact that a very symmetric decomposition of a manifold often is able to reveal deep properties of the manifold in a very striking and simple way.
This project is either about finding new symmetric decompositions using a clever algorithm (which you are welcome to make even smarter), studying known decomposition in more detail, or doing more theoretical work.
See also:
- Platonic solids
- Regular map (graph theory)
- Simplicial complexes
- Permutation group
- Hurwitz's automorphism theorem (automorphism is another word for "symmetry")
2. Graph encoded manifolds: It is quite easy to visualise orientable surfaces such as the sphere or the torus (the surface of a donut) embedded in three-dimensional space. For non-orientable surfaces, or manifolds of dimension higher than two this is a very challenging task. One very general way of achieving this (at least to a certain extent) is by representing a decomposition of a manifold by a graph with coloured edges.
A gem encoding a four-dimensional manifold.
Such graph encoded manifolds, or gems, can always be drawn on a sheet of paper while containing all the information about the manifold. While some of this information is very hard (or impossible) to access, some information can be read off the graph quite easily and other bits and pieces can be recovered by simple combinatorial rules.
This project is about using these simple combinatorial rules to deduce interesting facts about manifolds, to construct large families of such gems satisfying some properties (which is interesting for all kinds of reasons), to design a method to randomly generate such gems in certain settings (which is important for even more kinds of reasons), or to do more theoretical work.
See also:
- Lins, Sóstenes. Graph-encoded maps. J. Combin. Theory Ser. B 32 (1982), no. 2, 171–181.
- Lins, Sóstenes; Mandel, Arnaldo. Graph-encoded 3-manifolds. Discrete Math. 57 (1985), no. 3, 261–284.
These are research articles, for a simpler introduction, talk to me.
Timetable
Parameterised complexity for algorithmic problems in discrete geometry and topology:
Finding tractable algorithms to solve difficult topological problems for triangulations (of manifolds) which consist of an arbitrarily large number of pieces (the input size is unbounded), but are bounded with respect to some other quantity of the input. Examples of such quantities - or parameters - are first Betti number or the treewidth of the dual graph.
Finding such algorithms allows us to give an explanation for why difficult computational
problems in the field can often be solved efficiently in practice.
- Running times of an implemented fixed parameter tractable algorithm to compute Turaev-Viro invariants.
Tight triangulations:
Tightness is a generalisation of the notion of convexity which can be applied to triangulations of manifolds. After characterising tight triangulations in the previously largely unknown case of 3-dimensional manifolds, current efforts focus on explaining tightness in higher dimensions, where examples exhibit much more intricate topological features. The ultimate goal is to finally understand the framework as a whole.
- Tight and minimal triangulation of a real projective plane (shaded region is star of vertex 6): an icosahedron after antipodal identification.
3- and 4-manifold topology:
The core motivation of this theme is to provide discrete and computational methods to study manifolds. For this I switch between different types of triangulations which often provs to be highly effective.
One major part of this project is to describe minimal triangulations of manifolds. This project is joint with Stephan Tillmann and Hyam Rubinstein and has recently been funded by the ARC, see section on grants.
- The simply connected 4-manifold S2 x S2 encoded as an edge coloured graph.
Mathematical Software:
I am one of two lead developers of the computational topology software simpcomp. simpcomp is a
component of the well-known, open source computer algebra system GAP. The software allows the user to work with and analyse simplicial complexes.
See https://github.com/simpcomp-team/ for details.
| Project title | Research student |
|---|---|
| Topological Approaches to the Kashiwara Vergne Conjecture | Damian LIN |
| Complexity of 3-Manifolds | James MORGAN |
Publications
Book Chapters
- Rubinstein, H., Spreer, J., Tillmann, S. (2024). A New Family of Minimal Ideal Triangulations of Cusped Hyperbolic 3–manifolds. In David R Wood, Jan de Gier, Cheryl Praeger (Eds.), 2021-2022 Matrix Annals (Matrix Book #5), (pp. 5-24). Switzerland: Springer. [More Information]
- Spreer, J., Tillmann, S. (2018). Unravelling the Dodecahedral Spaces. In Jan de Gier, Cheryl E. Praeger, Terence Tao (Eds.), 2016 MATRIX Annals, (pp. 323-347). Cham: Springer International Publishing. [More Information]
- Burton, B., Maria, C., Spreer, J. (2015). Algorithms and Complexity for Turaev-Viro Invariants. In edited by Magnu´s M. Halldo´rsson, Kazuo Iwama, Naoki Kobayashi, Bettina Speckm (Eds.), Automata, Languages, and Programming : 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, (pp. 281-293). Heidelberg: Springer. [More Information]
Journals
- Jaco, W., Rubinstein, H., Spreer, J., Tillmann, S. (2025). Complexity of 3-manifolds obtained by Dehn filling. Algebraic and Geometric Topology, 25(1), 301-327. [More Information]
- Burton, B., Chang, H., Löffler, M., Maria, C., de Mesmay, A., Schleimer, S., Sedgwick, E., Spreer, J. (2024). Hard Diagrams of the Unknot. Experimental Mathematics, 33(3), 482-500. [More Information]
- Jaco, W., Rubinstein, H., Spreer, J., Tillmann, S. (2024). Slope norm and an algorithm to compute the crosscap number. Algebraic and Geometric Topology, 24(8), 4307-4351. [More Information]
Conferences
- Shankar, R., Spreer, J. (2023). A Uniform Sampling Procedure for Abstract Triangulations of Surfaces. 2023 SIAM Symposium on Algorithm Engineering and Experiments (ALENEX 2023), -: Society for Industrial and Applied Mathematics Publications. [More Information]
- Huszar, K., Spreer, J. (2019). 3-Manifold Triangulations with Small Treewidth. 35th International Symposium on Computational Geometry (SoCG 2019), Germany: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. [More Information]
- Bauer, U., Rathod, A., Spreer, J. (2019). Parametrized Complexity of Expansion Height. 27th Annual European Symposium on Algorithms (ESA 2019), Munich: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. [More Information]
2025
- Jaco, W., Rubinstein, H., Spreer, J., Tillmann, S. (2025). Complexity of 3-manifolds obtained by Dehn filling. Algebraic and Geometric Topology, 25(1), 301-327. [More Information]
2024
- Rubinstein, H., Spreer, J., Tillmann, S. (2024). A New Family of Minimal Ideal Triangulations of Cusped Hyperbolic 3–manifolds. In David R Wood, Jan de Gier, Cheryl Praeger (Eds.), 2021-2022 Matrix Annals (Matrix Book #5), (pp. 5-24). Switzerland: Springer. [More Information]
- Burton, B., Chang, H., Löffler, M., Maria, C., de Mesmay, A., Schleimer, S., Sedgwick, E., Spreer, J. (2024). Hard Diagrams of the Unknot. Experimental Mathematics, 33(3), 482-500. [More Information]
- Jaco, W., Rubinstein, H., Spreer, J., Tillmann, S. (2024). Slope norm and an algorithm to compute the crosscap number. Algebraic and Geometric Topology, 24(8), 4307-4351. [More Information]
2023
- Shankar, R., Spreer, J. (2023). A Uniform Sampling Procedure for Abstract Triangulations of Surfaces. 2023 SIAM Symposium on Algorithm Engineering and Experiments (ALENEX 2023), -: Society for Industrial and Applied Mathematics Publications. [More Information]
- Huszar, K., Spreer, J. (2023). On the Width of Complicated JSJ Decompositions. Leibniz International Proceedings in Informatics, 258(1 June 2023), 42-1-42-18. [More Information]
2022
- Spreer, J., Tillmann, S. (2022). Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations. Experimental Mathematics, 31(3), 897-907. [More Information]
- Burton, B., Datta, B., Spreer, J. (2022). Flip graphs of stacked and flag triangulations of the 2-sphere. The Electronic Journal of Combinatorics, 29(2), Paper 2.6-30 pages. [More Information]
2020
- Maria, C., Spreer, J. (2020). A Polynomial-Time Algorithm to Compute Turaev-Viro Invariants TV4,q of 3-Manifolds with Bounded First Betti Number. Foundations of Computational Mathematics, 20(5), 1013-1034. [More Information]
- Codenotti, G., Spreer, J., Santos, F. (2020). Average Betti numbers of induced subcomplexes in triangulations of manifolds. The Electronic Journal of Combinatorics, 27(3), 1-40. [More Information]
- Doolittle, J., Labbee, J., Lange, C., Sinn, R., Spreer, J., Ziegler, G. (2020). Combinatorial Inscridability Obstructions for Higher Dimensional Polytopes. Mathematika, 66(4), 927-953. [More Information]
2019
- Huszar, K., Spreer, J. (2019). 3-Manifold Triangulations with Small Treewidth. 35th International Symposium on Computational Geometry (SoCG 2019), Germany: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. [More Information]
- Huszar, K., Spreer, J., Wagner, U. (2019). On the treewidth of triangulated 3-manifolds. Journal of Computational Geometry, 10(2), 70-98. [More Information]
- Bauer, U., Rathod, A., Spreer, J. (2019). Parametrized Complexity of Expansion Height. 27th Annual European Symposium on Algorithms (ESA 2019), Munich: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. [More Information]
2018
- Burton, B., Datta, B., Singh, N., Spreer, J. (2018). A Construction Principle for Tight and Minimal Triangulations of Manifolds. Experimental Mathematics, 27(1), 22-36. [More Information]
- Burton, B., Maria, C., Spreer, J. (2018). Algorithms and complexity for Turaev-Viro invariants. Journal of Applied and Computational Topology, 2(44593), 33-53. [More Information]
- Huszar, K., Spreer, J., Wagner, U. (2018). On the Treewidth of Triangulated3-Manifolds. 34th International Symposium on Computational Geometry (SoCG 2018), Dagstuhl: Schloss Dagstuhl--Leibniz-Zentrum fur Informatik. [More Information]
2017
- Bagchi, B., Datta, B., Spreer, J. (2017). A characterization of tightly triangulated 3-manifolds. European Journal of Combinatorics, 61, 133-137. [More Information]
- Maria, C., Spreer, J. (2017). A polynomial time algorithm to compute quantum invariants of 3-manifolds with bounded first Betti number. SODA: 2017, New York: SIAM Publications. [More Information]
2016
- Spreer, J. (2016). A necessary condition for the tightness of odd-dimensional combinatorial manifolds. European Journal of Combinatorics, 51, 475-491. [More Information]
- Maria, C., Spreer, J. (2016). Admissible colourings of 3-manifold triangulations for Turaev-Viro type invariants. 24th Annual European Symposium on Algorithms (ESA 2016), Dagstuhl: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. [More Information]
- Jaco, W., Johnson, J., Spreer, J., Tillmann, S. (2016). Bounds for the genus of a normal surface. Geometry and Topology, 20(3), 1625-1671. [More Information]
2015
- Burton, B., Maria, C., Spreer, J. (2015). Algorithms and Complexity for Turaev-Viro Invariants. In edited by Magnu´s M. Halldo´rsson, Kazuo Iwama, Naoki Kobayashi, Bettina Speckm (Eds.), Automata, Languages, and Programming : 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, (pp. 281-293). Heidelberg: Springer. [More Information]
- Spreer, J. (2015). Random collapsibility and 3-sphere recognition. Computational Geometric and Algebraic Topology, Zurich: EMS Publishing House. [More Information]
- Burton, B., Datta, B., Singh, N., Spreer, J. (2015). Separation index of graphs and stacked 2-spheres. Journal of Combinatorial Theory, Series A, 136, 184-197. [More Information]
2014
- Spreer, J., Wagner, U. (2014). Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto. Workshop on Triangulations in Geometry and Topology, Japan: ----.
- Spreer, J. (2014). Combinatorial 3-Manifolds with Transitive Cyclic Symmetry. Discrete and Computational Geometry, 51(2), 394-426. [More Information]
2013
- Burton, B., Paixao, J., Spreer, J. (2013). Computational topology and normal surfaces: Theoretical and experimental complexity bounds. Meeting on Algorithm Engineering and Experiments (ALENEX 2013), New York City: SIAM Publications. [More Information]
- Burton, B., Spreer, J. (2013). Computationally proving triangulated 4-manifolds to be diffeomorphic. CG Week 2013 29th ACM Symposium on Computational Geometry + 2nd Computational Geometry: Applications, Practice, and Theory, New York: Association for Computing Machinery (ACM).
- Burton, B., Spreer, J. (2013). The complexity of detecting taut angle structures on triangulations. Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), New York City: SIAM Publications. [More Information]
2012
- Spreer, J. (2012). Partitioning the triangles of the cross polytope into surfaces. Beitraege zur Algebra und Geometrie (Contributions to Algebra and Geometry), 53(2), 473-486. [More Information]
2011
- Spreer, J., Kuhnel, W. (2011). Combinatorial properties of the K3 Surface: Simplicial blowups and slicings. Experimental Mathematics, 20(2), 201-216. [More Information]
- Spreer, J. (2011). Normal surfaces as combinatorial slicings. Discrete Mathematics, 311(14), 1295-1309. [More Information]
- Effenberger, F., Spreer, J. (2011). Simplicial blowups and discrete normal surfaces in simpcomp. ACM Communications in Computer Algebra, 45(3-4), 173-176. [More Information]
2010
- Effenberger, F., Spreer, J. (2010). simpcomp - a GAP toolbox for simplicial complexes. ACM Communications in Computer Algebra, 44, 186-189. [More Information]
Selected Grants
2022
- Faculty of Science Start up funding - Triangulations - Jonathan Spreer, Spreer J, Faculty of Science/Faculty Startup Scheme
- Triangulations: linking geometry and topology with combinatorics, Spreer J, Australian Research Council (ARC)/Discovery Projects (DP)
2019
- Trisections, triangulations and the complexity of manifolds, Tillmann S, Rubinstein H, Spreer J, Australian Research Council (ARC)/Discovery Projects (DP)