Martin Winter, PhD
martin.winter@mis.mpg.de

I am a mathematician. I am member of the Nonlinear Algebra group at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig and principal investigator in the priority program Combinatorial Synergies.

My research is in discrete geometry, real algebraic geometry, geometric rigidity, semi-definite optimization, spectral graph theory, algebraic combinatorics and low-dimensional topology. Most recently I am studying the Wachspress Geometry of convex polytopes. I am also leading a Lean formalization project about polyhedral geometry and combinatorics.

Until 2025 I was Dirichlet fellow at TU Berlin. Before I was postdoc at the University of Warwick (UK). I got my PhD from TU Chemnitz (Germany) in 2021.

What’s new

• I am currently at the Workshop Discrete Geometric Structures 2026 at TU Wien (August 3 – 7, 2026). On Friday I will give a short talk on how to not worry about non-convexity.
• New preprint. “Shallower ReLU Network Representations via Exact Linear Algebra” (with Kilian Rueß, Gennadiy Averkov, Florestan Brunck, Moritz Grillo, Christoph Hertrich, Georg Loho, Jack Stade, Moritz Stargalla and Matthew Sun). We prove that a ReLU network can represent the maximum of 10 real numbers with onyl two hidden layers.
• New preprint. “Second-order rigidity of coned polytope frameworks and the stress-flex conjecture from a vector-valued Schläfli formula” (with Eleni Pachyli and Roman Prosanov). We derive the weak stress-flex conjecture (for the Wachspress stress) from a vector valued Schläfli-type formula, for which we give a new discrete-geometric proof. From this we prove that coned polytope frameworks are prestress stable.
• New preprint. “Deformations and second-order rigidity of polytopes” (with Matthias Adrian-Himmelmann and Zhen Zhang). We develop a second-order theory for polytope rigidity and use it to decide the rigidity of previously open cases.
• Georg Loho, Olivia Röhrig and I are organizing a 2-weeks hands-on workshop “Polyhedra in Lean” to take place August 24 – September 4, 2026 at FU Berlin.

What’s coming

• I organize the workshop “Polyhedra in Lean” at FU Berlin (August 24 – September 4, 2026).
• I will be at the 2026 Annual Conference of the “Combinatorial Synergies” Priority Program at the Goethe University Franfkurt (September 14 – 16, 2026)
• I will participate in the Conference “Positive Geometry Beyond Physics” at MPI Leipzig (October 21 – 23, 2026).

Selected publications

• The canonical form, scissors congruence and adjoint degrees of polytopes [arXiv]
  with Tom Baumbach, Ansgar Freyer and Julian Weigert
• Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints [arXiv, journal]
  This paper won the FSEM Post-Doctoral Research Prize 2024 from the University of Warwick
• Kalai’s \(3^d\) conjecture for unconditional and locally anti-blocking polytopes [arXiv, journal]
  with Raman Sanyal
• “Rigidity of polytopes with edge length and coplanarity constraints” [arXiv, journal]
  with Matthias Adrian-Himmelmann and Bernd Schulze

I am an active user of the StackExchange network. Here are links to my mathematical profiles:

• MathOverflow
• Math.StackExchange