Computational Mathematics

computer-assistance in proofs; flag algebras in combinatorics; algebraic methods in computation; interactive theorem provers; learning-based heuristics

Exploring optimization, learning, and formal proof verification to advance pure mathematics.

What we are interested in

Computers and Artificial Intelligence have always been an important tool for mathematicians, allowing one to gather data and extrapolate connections to formulate conjectures, exhaustively execute case analysis too large to be done by hande, solve underlying optimization problems, or to formalize and verify proofs. We are interested in exploring the many ways in which computational tools can be used to advance mathematics and formulate novel results by leveraging our unique knowledge of and access to the ZIB’s computational resources.

Members

Christoph Spiegel
spiegel (at) zib.de

Aldo Kiem
kiem (at) zib.de

Konrad Mundinger
mundinger (at) zib.de

Lídia Rossell Rodríguez
lidia.rossell (at) estudiantat.upc.edu

Max Zimmer
zimmer (at) zib.de

Yves Jäckle
jaeckle (at) zib.de

Projects

Learning Extremal Structures in Combinatorics (MATH+ EF1-12)
Scaling Up Flag Algebras in Combinatorics (MATH+ EF1-21)
LEAN on Me: Transforming Mathematics Through Formal Verification, Improved Tactics, and Machine Learning (MATH+ AA5-9)

Publications

Mundinger, K., Pokutta, S., Spiegel, C., and Zimmer, M. (2024). Extending the Continuum of Six-Colorings. [arXiv]

[BibTeX]

@misc{mpsz_hadwigernelsonspectrum_24,
  archiveprefix = {arXiv},
  eprint = {2404.05509},
  year = {2024},
  author = {Mundinger, Konrad and Pokutta, Sebastian and Spiegel, Christoph and Zimmer, Max},
  title = {Extending the Continuum of Six-Colorings}
}

Kiem, A., Pokutta, S., and Spiegel, C. (2023). The 4-color Ramsey Multiplicity of Triangles. [arXiv] [code]

[BibTeX]

@misc{kps_flagalgebrasymmetries_22,
  archiveprefix = {arXiv},
  eprint = {2312.08049},
  primaryclass = {math.CO},
  year = {2023},
  author = {Kiem, Aldo and Pokutta, Sebastian and Spiegel, Christoph},
  title = {The 4-color Ramsey Multiplicity of Triangles},
  code = {https://github.com/FordUniver/kps_trianglemult}
}

Parczyk, O., Pokutta, S., Spiegel, C., and Szabó, T. (2023). Fully Computer-assisted Proofs in Extremal Combinatorics. Proceedings of AAAI Conference on Artificial Intelligence. [arXiv] [slides] [code]

[BibTeX]

@inproceedings{ppss_ramsey_22,
  year = {2023},
  booktitle = {Proceedings of AAAI Conference on Artificial Intelligence},
  archiveprefix = {arXiv},
  eprint = {2206.04036},
  primaryclass = {math.CO},
  author = {Parczyk, Olaf and Pokutta, Sebastian and Spiegel, Christoph and Szabó, Tibor},
  title = {Fully Computer-assisted Proofs in Extremal Combinatorics},
  code = {https://zenodo.org/record/6602512#.YyvFhi8Rr5g},
  slides = {https://pokutta.com/slides/20220705_DMD22_RamseyMultiplicity.pdf}
}

Rué Perna, J. J., and Spiegel, C. (2023). The Rado Multiplicity Problem in Vector Spaces Over Finite Fields. Proceedings of European Conference on Combinatorics. [arXiv] [code]

[BibTeX]

@inproceedings{rs_radomult_23,
  year = {2023},
  booktitle = {Proceedings of European Conference on Combinatorics},
  archiveprefix = {arXiv},
  eprint = {2304.00400},
  primaryclass = {math.CO},
  author = {Rué Perna, Juan José and Spiegel, Christoph},
  title = {The Rado Multiplicity Problem in Vector Spaces Over Finite Fields},
  code = {https://github.com/FordUniver/rs_radomult_23}
}

Braun, G., Pokutta, S., and Weismantel, R. (2022). Alternating Linear Minimization: Revisiting von Neumann’s Alternating Projections. [arXiv] [slides] [video]

[BibTeX]

@misc{alternating_lmo_2022,
  archiveprefix = {arXiv},
  eprint = {2212.02933},
  primaryclass = {math.OC},
  year = {2022},
  author = {Braun, Gábor and Pokutta, Sebastian and Weismantel, Robert},
  title = {Alternating Linear Minimization: Revisiting von Neumann’s Alternating Projections},
  slides = {https://pokutta.com/slides/20230327-icerm.pdf},
  video = {https://icerm.brown.edu/programs/sp-s23/w2/#schedule-item-4945}
}

Kamčev, N., and Spiegel, C. (2022). Another Note on Intervals in the Hales-Jewett Theorem. Electronic Journal of Combinatorics, 29(1). DOI: 10.37236/9400 [URL] [arXiv]

[BibTeX]

@article{ks_halesjewett_18,
  year = {2022},
  journal = {Electronic Journal of Combinatorics},
  volume = {29},
  number = {1},
  doi = {10.37236/9400},
  url = {https://combinatorics.org/ojs/index.php/eljc/article/view/v29i1p62},
  archiveprefix = {arXiv},
  eprint = {1811.04628},
  primaryclass = {math.CO},
  author = {Kamčev, Nina and Spiegel, Christoph},
  title = {Another Note on Intervals in the Hales-Jewett Theorem}
}

Cao-Labora, G., Rué Perna, J. J., and Spiegel, C. (2021-10). An Erdős-Fuchs Theorem for Ordered Representation Functions. Ramanujan Journal, 56, 183–2091. DOI: 10.1007/s11139-020-00326-2 [URL] [arXiv]

[BibTeX]

@article{crs_erdosfuchs_19,
  year = {2021-10},
  journal = {Ramanujan Journal},
  volume = {56},
  pages = {183-2091},
  doi = {10.1007/s11139-020-00326-2},
  url = {https://link.springer.com/article/10.1007/s11139-020-00326-2},
  archiveprefix = {arXiv},
  eprint = {1911.12313},
  primaryclass = {math.NT},
  author = {Cao-Labora, Gonzalo and Rué Perna, Juan José and Spiegel, Christoph},
  title = {An Erdős-Fuchs Theorem for Ordered Representation Functions}
}

Fabian, D., Rué Perna, J. J., and Spiegel, C. (2021-08). On Strong Infinite Sidon and Bₕ Sets and Random Sets of Integers. Journal of Combinatorial Theory, Series A, 182. DOI: 10.1016/j.jcta.2021.105460 [URL] [arXiv]

[BibTeX]

@article{frs_sidon_19,
  year = {2021-08},
  journal = {Journal of Combinatorial Theory, Series A},
  volume = {182},
  doi = {10.1016/j.jcta.2021.105460},
  url = {https://sciencedirect.com/science/article/abs/pii/S0097316521000595},
  archiveprefix = {arXiv},
  eprint = {1911.13275},
  primaryclass = {math.CO},
  author = {Fabian, David and Rué Perna, Juan José and Spiegel, Christoph},
  title = {On Strong Infinite Sidon and Bₕ Sets and Random Sets of Integers}
}

Corsten, J., Mond, A., Pokrovskiy, A., Spiegel, C., and Szabó, T. (2020-10). On the Odd Cycle Game and Connected Rules. European Journal of Combinatorics, 89. DOI: 10.1016/j.ejc.2020.103140 [URL] [arXiv]

[BibTeX]

@article{cmpss_oddcyclegame_19,
  year = {2020-10},
  journal = {European Journal of Combinatorics},
  volume = {89},
  doi = {10.1016/j.ejc.2020.103140},
  url = {https://sciencedirect.com/science/article/abs/pii/S0195669820300615},
  archiveprefix = {arXiv},
  eprint = {1906.04024},
  primaryclass = {math.CO},
  author = {Corsten, Jan and Mond, Adva and Pokrovskiy, Alexey and Spiegel, Christoph and Szabó, Tibor},
  title = {On the Odd Cycle Game and Connected Rules}
}

Candela, P., Serra, O., and Spiegel, C. (2020). A Step Beyond Freĭman’s Theorem for Set Addition Modulo a Prime. Journal De Théorie Des Nombres De Bordeaux, 32(1), 275–289. DOI: 10.5802/jtnb.1122 [URL] [arXiv]

[BibTeX]

@article{css_freiman_18,
  year = {2020},
  journal = {Journal de Théorie des Nombres de Bordeaux},
  volume = {32},
  number = {1},
  pages = {275-289},
  doi = {10.5802/jtnb.1122},
  url = {https://jtnb.centre-mersenne.org/item/JTNB_2020__32_1_275_0/},
  archiveprefix = {arXiv},
  eprint = {1805.12374},
  primaryclass = {math.CO},
  author = {Candela, Pablo and Serra, Oriol and Spiegel, Christoph},
  title = {A Step Beyond Freĭman's Theorem for Set Addition Modulo a Prime}
}

Rué Perna, J. J., and Spiegel, C. (2020). On a Problem of Sárközy and Sós for Multivariate Linear Forms. Revista Matemática Iberoamericana, 36(7), 2107–2119. DOI: 10.4171/RMI/1193 [URL] [arXiv]

[BibTeX]

@article{rs_sarkozysos_18,
  year = {2020},
  journal = {Revista Matemática Iberoamericana},
  volume = {36},
  number = {7},
  pages = {2107-2119},
  doi = {10.4171/RMI/1193},
  url = {https://ems.press/journals/rmi/articles/16818},
  archiveprefix = {arXiv},
  eprint = {1802.07597},
  primaryclass = {math.CO},
  author = {Rué Perna, Juan José and Spiegel, Christoph},
  title = {On a Problem of Sárközy and Sós for Multivariate Linear Forms}
}

Kusch, C., Rué Perna, J. J., Spiegel, C., and Szabó, T. (2019-09). On the Optimality of the Uniform Random Strategy. Random Structures & Algorithms, 55(2), 371–401. DOI: 10.1002/rsa.20829 [URL] [arXiv]

[BibTeX]

@article{krss_randomstrategy_17,
  year = {2019-09},
  journal = {Random Structures & Algorithms},
  volume = {55},
  number = {2},
  pages = {371-401},
  doi = {10.1002/rsa.20829},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20829},
  archiveprefix = {arXiv},
  eprint = {1711.07251},
  primaryclass = {math.CO},
  author = {Kusch, Christopher and Rué Perna, Juan José and Spiegel, Christoph and Szabó, Tibor},
  title = {On the Optimality of the Uniform Random Strategy}
}

Freĭman, G. A., Serra, O., and Spiegel, C. (2019). Additive Volume of Sets Contained in Few Arithmetic Progressions. INTEGERS, 19. [URL] [arXiv]

[BibTeX]

@article{fss_additivevolume_18,
  year = {2019},
  journal = {INTEGERS},
  volume = {19},
  url = {https://math.colgate.edu/~integers/t34/t34.mail.html},
  archiveprefix = {arXiv},
  eprint = {1808.08455},
  primaryclass = {math.NT},
  author = {Freĭman, Gregory A. and Serra, Oriol and Spiegel, Christoph},
  title = {Additive Volume of Sets Contained in Few Arithmetic Progressions}
}

Salia, N., Spiegel, C., Tompkins, C., and Zamora, O. (2019). Independent Chains in Acyclic Posets. [arXiv]

[BibTeX]

@misc{sstz_poset_19,
  archiveprefix = {arXiv},
  eprint = {1912.03288},
  primaryclass = {math.CO},
  year = {2019},
  author = {Salia, Nika and Spiegel, Christoph and Tompkins, Casey and Zamora, Oscar},
  title = {Independent Chains in Acyclic Posets}
}

Rué Perna, J. J., Spiegel, C., and Zumalacárregui, A. (2018). Threshold Functions and Poisson Convergence for Systems of Equations in Random Sets. Mathematische Zeitschrift, 288, 333–360. DOI: 10.1007/s00209-017-1891-2 [URL] [arXiv]

[BibTeX]

@article{rsz_threshold_12,
  year = {2018},
  journal = {Mathematische Zeitschrift},
  month = feb,
  volume = {288},
  pages = {333-360},
  doi = {10.1007/s00209-017-1891-2},
  url = {https://link.springer.com/article/10.1007/s00209-017-1891-2},
  archiveprefix = {arXiv},
  eprint = {1212.5496},
  primaryclass = {math.CO},
  author = {Rué Perna, Juan José and Spiegel, Christoph and Zumalacárregui, Ana},
  title = {Threshold Functions and Poisson Convergence for Systems of Equations in Random Sets}
}

Spiegel, C. (2017). A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems. Electronic Journal of Combinatorics, 24(3). DOI: 10.37236/6730 [URL] [arXiv]

[BibTeX]

@article{spiegel_supersaturation_17,
  year = {2017},
  journal = {Electronic Journal of Combinatorics},
  volume = {24},
  number = {3},
  doi = {10.37236/6730},
  url = {https://combinatorics.org/ojs/index.php/eljc/article/view/v24i3p38},
  archiveprefix = {arXiv},
  eprint = {1701.01631},
  primaryclass = {math.CO},
  author = {Spiegel, Christoph},
  title = {A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems}
}