Skip to main content Skip to navigation

Dr Martin Winter

  A picture of me

Dr Martin Winter

Research Fellow
Office: B2.28
Phone: +44 (0)24 765 23972

My favourite polyhedron

I am a postdoc working with Agelos Georgakopoulos funded by the EPSRC grant "Graph Theory in Higher Dimensions". Starting in July I will be Dirichlet Fellow at TU Berlin. I am also PI in the priority program "Combinatorial Synergies" by the German Research Foundation with my project "Wachspress Coordinates - a Bridge betwen Algebra, Geometry and Combinatorics". From April 2025 I will conduct this project at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig (Germany).

My main mathematical interest is Geometry in its broadest sense, including discrete, algebraic, convex and computational, also touching on many aspects of graph theory, topology and representation theory. I work(ed) and published on

  • polyhedral combinatorics and geometry,
  • graph theory (spectral, algebraic, geometric and topological),
  • rigidity theory, geometric constraint system and real algebraic geometry,
  • low-dimensional topology (knot theory, embeddings and embeddability of graphs and complexes, ...)
  • symmetries of discrete and geometric objects, finite group theory and real representation theory,
  • algebraic combinatorics (association schemes, distance regular graphs, codes ...),
  • convex geometry,
  • random structures and graph limits,
  • discrete dynamical systems, ...

Where you can meet me next ...

Publications & Preprints
  • Vertex-Facet Assignments for Polytopes (open access; arXiv)
    with Thomas Jahn
    Contributions to Algebra and Geometry (2020).
  • Geometry and Topology of Symmetric Point Arrangements (link; arXiv)
    Linear Algebra and its Applications (2021),
  • Classification of Vertex-Transitive Zonotopes (open access; pdf; arXiv)
    Discrete & Computational Geometry (2021),
  • The Edge-Transitive Polytopes that are not Vertex-Transitive (link; arXiv)
    with Frank Göring
    Ars Mathematica Contemporanea (2022), doi:10.26493/1855-3974.2712.6be
  • Symmetric and Spectral Realizations of Highly Symmetric Graphs (arXiv)
  • Eigenpolytopes, Spectral Polytopes and Edge-Transitivity (arXiv)
  • Capturing Polytopal Symmetries by Coloring the Edge-Graph (open access; arXiv)
    Discrete & Computational Geometry (2023),
  • (Random) Trees of Intermediate Volume Growth (arXiv, extended abstract)
    with George Kontogeorgiou
    Extended abstract published at Eurocomb'23
  • Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints (open access; arXiv)
    International Mathematics Research Notices (2023),
    This paper won the FSEM Post-Doctoral Research Prize 2024 from the University of Warwick
  • Characterizing Clique Convergence for Locally Cyclic Graphs of Minimum Degree d>=6 (arXiv)
    with Anna M. Limbach
  • The clique graphs of the hexagonal lattice - an explicit construction and a short proof of divergence (open access; arXiv)
    Discrete Mathematics (2024),
  • Kalai's 3^d conjecture for unconditional and locally anti-blocking polytopes (arXiv)
    with Raman Sanyal
    accepted at Proceedings of the AMS
  • The stress-flex conjecture (arXiv)
    with Robert Connelly, Steven J. Gortler, Louis Theran
Conferences, Workshops, etc.


  • South Eastern Graph Theory Workshop (Chemnitz)
  • FRICO (Chemnitz), Talk: "Spectral Methods for Symmetric Polytopes"
  • South Eastern Graph Theory Workshop (Freiberg)
  • Geomtry Day (Dresden)


  • Polytope Spring School (Bochum), Talk: "Spectral Methods for Symmetric Polytopes"
  • Summer School on Geometric & Algebraic Combinatorics (GAC, Paris)
  • Discrete Geometry Day^2 (Budapest), Talk: "Spectral Methods for Symmetric Polytopes" (slides)
  • South Eastern Graph Theory Workshop (Chemnitz), Talk: "Edge-Transitive Polytopes"
  • Colloquium on Combiantorics (Kolkom, Paderborn), Talk: "Edge-Transitive Polytopes" (slides)
  • Geometry Day (Jena), Talk: "Edge-Transitive Polytopes"


  • Combinatorial Coworkspace (Kleinwalsertal), Talk: "Inscribed Zonotopes and Hyperplane Arrangements with Congruent Chambers" (slides)
  • DMV Annual Conference (Chemnitz), Talk: "Spectral Graph Theory for Polytopes" (slides)


  • (Polytop)ics (MPI Leipzig, online), Talk: "Capturing Polytopal Symmetries in the Edge-Graph" (slides, video)
  • DMV-ÖMG Annual Conference (Passau)




Further Material


  • "Capturing Polytopal Symmetries in the Edge-Graph" (extended slides) from my talk at KTH (Stockholm, Sweden)
  • "On Symmetry, Rigidity and Spectrum" (slides) from my talk at the "School of Mathematical & Statistical Science" at NUI Galway (Ireland)
  • "Some Modern Open Problems on Convex Polytopes with Symmetries and Regularities" (slides) from my talk at the "Combinatorics Seminar" at Warwick (UK)
  • "When Edges shrink then Polytopes become smaller - a proof in geometric polytope theory" (slides) from my talk at Pretoria University (South Africa)
  • "(Random) Trees of Intermediate Volume Growth" (slides) from my talk in the Algorithms and Complexity seminar at Durham University
  • "On 4-Flat Graphs and the Embeddability of 2-Complexes in Dimension Four" (slides) from my talk in the Discrete Geometry seminar at FU Berlin
  • "Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints" (slides) from my talk at the Seminar "Diskrete Mathematik/Geometrie & Diskrete Geometrie" at TU Berlin
  • "Wachspress Objects and the Reconstruction of Convex Polytopes from Metric Data" (slides) from my talk at the Joint Algebra Seminar at RWTH Aachen
  • "Wachspress Objects and the Reconstruction of Convex Polytopes from Metric Data" (slides) from my talk at the Algebra & Geometry Seminar at KTH (Stockholm)

PhD Thesis

My Thesis "Spectral Realizations of Symmetric Graphs, Spectral Polytopes and Edge-Transitivity" and the slides of my defense talk.


In the winter term 2022 I held a lecture on "Polytope Theory" as part of the Taught Course Center at Bath, Bristol, Imperial, Oxford, Warwick and Swansea

I am an active user of the StackExchange network (username: M. Winter).
These are links to my mathematical profiles:

I am also an active contributor to Wikipedia.