NeuCurves MATH+ EF-LiOpt-1 ongoing

Neural Generative Models for Algebraic Curves

This project aims to apply neural generative models to real plane algebraic curves to advance the classification of M-curves, a problem from Hilbert's 16th problem that remains open for degree ≥8.

Sep 2025 to Aug 2028Sep 25 to Aug 28
5
2

IOL Project Members

Sebastian PokuttaTU/PR/Dominic Simon
Sebastian Pokutta Principal Investigator
Christoph Spiegel
Christoph Spiegel Project Member
Konrad Plato
Konrad Plato Project Member
Max Zimmer
Max Zimmer Project Member

External Project Members

Michael Joswig
Michael Joswig Principal Investigator

Funding

This project is being funded by the Berlin Mathematics Research Center MATH+ (project ID EF-LiOpt-1), itself funded by the German Research Foundation (DFG) under Germany's Excellence Strategy (EXC-2046/1, project ID 390685689) from September 2025 to August 2028.

Project Description

Harnack (1876) showed that a real plane curve of degree d has at most ½(d²−3d+4) ovals; curves achieving this bound are M-curves. Classifying M-curves by isotopy type—the first part of Hilbert’s 16th problem—remains open for d≥8.

This project aims to develop neural generative models for the triangulations and sign distributions used in Viro’s combinatorial patchworking, a technique that encodes curve isotopy types and underlies tropical geometry. Complete enumeration is infeasible and random sampling ineffective at relevant scales, making learned generative approaches essential.

Talks and posters

Conference and workshop talks

Jul 2026
Fast Isotopy Computation for $T$-Curves by Christoph Spiegel
8th DOxML Conference, Tokyo [PDF]

Publications and preprints

Preprints

  1. Geiselmann, Z., Joswig, M., Kastner, L., Mundinger, K., Pokutta, S., Spiegel, C., Wack, M., and Zimmer, M. (2026). Fast Isotopy Computation for T-Curves. [arXiv]
    [BibTeX]
    @misc{2026_ZoeEtAl_Isotopytcurves_2604-09221,
  archiveprefix = {arXiv},
  eprint = {2604.09221},
  arxiv = {arXiv:2604.09221},
  primaryclass = {math.AG},
  year = {2026},
  author = {Geiselmann, Zoe and Joswig, Michael and Kastner, Lars and Mundinger, Konrad and Pokutta, Sebastian and Spiegel, Christoph and Wack, Marcel and Zimmer, Max},
  title = {Fast Isotopy Computation for T-Curves},
  date = {2026-04-10}
}
  2. Geiselmann, Z., Joswig, M., Kastner, L., Mundinger, K., Pokutta, S., Spiegel, C., Wack, M., and Zimmer, M. (2026). Limits of combinatorial patchworking. [arXiv]
    [BibTeX]
    @misc{2026_ZoeEtAl_Patchworkedcurves_2602-06888,
  archiveprefix = {arXiv},
  eprint = {2602.06888},
  arxiv = {arXiv:2602.06888},
  primaryclass = {math.AG},
  year = {2026},
  author = {Geiselmann, Zoe and Joswig, Michael and Kastner, Lars and Mundinger, Konrad and Pokutta, Sebastian and Spiegel, Christoph and Wack, Marcel and Zimmer, Max},
  title = {Limits of combinatorial patchworking},
  date = {2026-02-06}
}