Web Links

http://www.maths.lancs.ac.uk/~schulzeb/

Current Research

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

PhD Supervision Interests

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).