Dr Jonathan Spreer
F07  Carslaw Building
The University of Sydney


Website 
Personal Website My software simpcomp Photo by Kay Herschelmann. 
Biographical details
since 2019 
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 AustraliaIndia Strategic Research Fund (AISRF). Grant AISRF06660, at the School of Mathematics and Physics, University of Queensland.  
20142015 
Postdoctoral Research Fellow, funded by the AustraliaIndia Strategic Research Fund (AISRF). Grant AISRF06660, at the School of Mathematics and Physics, University of Queensland.  
20122013  Postdoctoral Research Fellow at the School of Mathematics and Physics, University of Queensland.  
 
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 4manifolds, in particular the K3Surface (Supervisor: Prof.W.Kühnel).  
2007  Maîtrise (french one year postgraduate diploma) in mathematics at the Université Pierre et Marie Curie in Paris. 
Research interests
My research interests are motivated by the study of manifolds (i.e., surfaces and their higherdimensional analogues). For this, I use a blend of techniques from mathematical fields such as lowdimensional 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.
Teaching and supervision
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, ngons), 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 wellknown 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 threedimensional space. For nonorientable 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 fourdimensional 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. Graphencoded maps. J. Combin. Theory Ser. B 32 (1982), no. 2, 171–181.
 Lins, Sóstenes; Mandel, Arnaldo. Graphencoded 3manifolds. Discrete Math. 57 (1985), no. 3, 261–284.
These are research articles, for a simpler introduction, talk to me.
Timetable
J_Spreer
Current projects
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 TuraevViro 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 3dimensional 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 4manifold 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 4manifold S^{2} x S^{2} 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 wellknown, open source computer algebra system GAP. The software allows the user to work with and analyse simplicial complexes.
See https://github.com/simpcompteam/ for details.
Selected grants
2019
 Trisections, triangulations and the complexity of manifolds; Tillmann S, Rubinstein H, Spreer J; Australian Research Council (ARC)/Discovery Projects (DP).
Selected publications





















