Applied Discrete Geometry
Rigidity and flexibility of discrete geometric constraint systems and applications
Combinatorial Optimization
Graph Theory
Algebraic methods in discrete mathematics
I would be happy to supervise a PhD student who is interested in graph theory, combinatorics and/or discrete geometry (in particular rigidity and flexibility of geometric constraint systems, geometric graphs, count matroids, and symmetry in discrete structures). I have purely mathematical projects as well as projects that are motivated by applications and are interdisciplinary (involving areas such as robotics, structural and mechanical engineering, biophysics, and materials science).
I lead the research theme Geometric Rigidity Theory at Lancaster University. I am also part of the Combinatorics and Algebra & Geometry research groups.
A lot of my recent work has been motivated by problems from industry and the applied sciences, and hence a significant part of my research is interdisciplinary. A particular focus of my research lies in the field of Geometric Rigidity Theory, which is concerned with the rigidity and flexibility of discrete structures that are defined by geometric constraints (fixed lengths, directions, angles, etc.) on a set of primitive geometries (points, line segments, rigid bodies, etc.). I work on both combinatorial and geometric problems in this area. Recently I have also made connections to areas such as topological graph theory, algebraic topology (cosheaf homology) and lattice geometry. Collaborations with applied scientists and industry professionals include ongoing work with structural engineers such as William Baker (SOM), Cameron Millar (SOM), Simon Guest (Cambridge), and Allan McRobie (Cambridge), chemists such as Patrick Fowler (Sheffield), and formation control theorists, such as Daniel Zelazo (Technion).
Submitted papers:
Matthias Himmelmann, Bernd Schulze and Martin Winter, Rigidity of polytopes with edge length and coplanarity constraints, arxiv.org/abs/2505.00874
Leah Berman, Signe Lundqvist, Bernd Schulze, Brigitte Servatius, Herman Servatius, Klara Stokes, Walter Whiteley, Counting for rigidity under projective transformations in the plane, arxiv.org/abs/2503.07228
Cameron Millar, Bernd Schulze, and Louis Theran, Equilibirum stresses in frameworks via symmetric averaging, arxiv.org/abs/2502.15410
Jack Esson, Eleftherios Kastis and Bernd Schulze, Orientation-Reversing Crystallographic Rigidity, arxiv.org/abs/2502.14392v1
Oleg Karpenkov, Fatemeh Mohammadi, Christian Mueller and Bernd Schulze, Klein-Arnold tensegrities, arxiv.org/abs/2410.12729#
Alison LaPorta and Bernd Schulze, Generic infinitesimal rigidity for rotational groups in the plane, arxiv.org/abs/2410.07931
Leah Berman, Signe Lundqvist, Bernd Schulze, Brigitte Servatius, Herman Servatius, Klara Stokes, and Walter Whiteley, Projective rigidity of point-line configurations in the plane, arxiv.org/abs/2407.17836
Alison LaPorta and Bernd Schulze, Rigidity of symmetric frameworks with non-free group actions on the vertices, arxiv.org/abs/2407.13612
Oleg Karpenkov, Fatemeh Mohammadi, Christian Mueller and Bernd Schulze, A differential approach to Maxwell-Cremona liftings, arxiv.org/abs/2312.09891
James Cruickshank and Bernd Schulze, Symmetric contact systems of segments, pseudotriangulations and inductive constructions for corresponding surface graphs, arxiv.org/abs/2006.10519
PhD students:
Jack Esson (2023 - )
Alison La Porta (2021 - 2025)
Josph Wall (2019 - 2024)
Hattie Serocold (2014 - 2018)
Conference and workshop organisation:
Workshop on Theory and Algorithms in Graph Rigidity and Algebraic Statistics, NII Shonan, Kanagawa, Japan, Sept. 2024
ICMS Knowledge Exchange Workshop on Geometric Rigidity, Graphic Statics and Engineering, SOM London Office, London, UK, June 2024
Mini-symposium on matroids and combinatorial geometry, 29th British Combinatorial Conference, Lancaster University, UK, July 2022
Workshop on Geometric constraint systems: rigidity, flexibility and applications, Lancaster University, UK, June 2019
Workshop on Bond-node structures: rigidity, combinatorics and Chemistry, Lancaster University, UK, June 2018
Workshop on Bond-node structures: rigidity, combinatorics and Materials Science, Lancaster University, UK, June 2017
Workshop on Geometric Rigidity Theory and Applications, International Centre for Mathematical Sciences (ICMS), Edinburgh, UK, May 2016
Workshop on Making models: Stimulating research in rigidity theory and spatial-visual reasoning, Fields Institute, Toronto, Canada, Aug. 2014
