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  "author_context_policy": "Optional context is written by the author, not an independent evaluation. These public records are equally accessible to people and automated clients.",
  "papers": [
    {
      "id": "barany-larman-primes",
      "title": "The topological Bárány–Larman conjecture for prime numbers",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "preprint",
      "type": "research",
      "reference": "",
      "topics": [
        "tverberg",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/barany-larman-primes/",
      "arxiv": "2609.37876",
      "doi": null,
      "pdf": null,
      "abstract": "The Bárány–Larman conjecture states that for any d+1 sets of r points each in ℝᵈ, considered as color classes, we can partition their union into r rainbow (d+1)-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when r is a prime number. We also show that the optimal colorful Tverberg theorem of Blagojević, Matschke, and Ziegler cannot be extended to prime powers.",
      "summary": "The topological Bárány–Larman conjecture for prime numbers, and counterexamples to extending the optimal colorful Tverberg theorem to prime powers.",
      "methods": [],
      "related_problems": [],
      "resources": [
        {
          "label": "Verification files on arXiv",
          "url": "https://arxiv.org/abs/2609.37876#anc"
        }
      ],
      "videos": [],
      "date": "2026-09-29"
    },
    {
      "id": "discrepancy-tverberg-regression",
      "title": "Discrepancy theory, Tverberg's theorem, and regression depth",
      "authors": [
        "Aleksey Lopez",
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "preprint",
      "type": "research",
      "reference": "",
      "topics": [
        "tverberg",
        "discrepancy"
      ],
      "student_authors": [
        "Aleksey Lopez"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/discrepancy-tverberg-regression/",
      "arxiv": "2609.24775",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "Discrepancy-theoretic bounds for Tverberg partitions with tolerance, for point sets and for regression hulls of hyperplanes.",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": [],
      "date": "2026-09-21"
    },
    {
      "id": "helly-radon-flats",
      "title": "Helly and Radon theorems for convex intersections containing k-flats",
      "authors": [
        "Sarah Ludwigson",
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "preprint",
      "type": "research",
      "reference": "",
      "topics": [
        "helly",
        "tverberg",
        "topological-combinatorics"
      ],
      "student_authors": [
        "Sarah Ludwigson"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/helly-radon-flats/",
      "arxiv": "2608.27189",
      "doi": null,
      "pdf": null,
      "abstract": "We study versions of results in combinatorial geometry related to families of convex sets in ℝᵈ whose intersection contains a k-dimensional affine space. We prove generalizations of the colorful Radon theorem, the fractional Helly theorem, the colorful Helly theorem, and the selection-structure Helly theorem. When k=0, our arguments give new proofs of the corresponding versions for points.",
      "summary": "Colorful and fractional intersection theorems for convex families whose intersection contains an affine k-flat.",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": [],
      "date": "2026-08-27"
    },
    {
      "id": "four-hyperplanes",
      "title": "Four hyperplanes do not always equipartition a mass in ℝ⁴",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "mass-partitions"
      ],
      "student_authors": [],
      "featured": true,
      "url": "https://www.psoberon.com/papers/four-hyperplanes/",
      "arxiv": "2608.23312",
      "doi": null,
      "pdf": null,
      "abstract": "We construct a smooth strictly positive density in ℝ⁴ that cannot be divided into 16 parts of the same size by four affine hyperplanes. This settles the last open case of Grünbaum's 1960 conjecture and disproves Ramos' general conjecture on hyperplane equipartitions.\n\nWe reduce the construction to finding two homogeneous polynomials in four variables, of degrees three and four, whose multilinear coefficients cannot vanish simultaneously after any orthogonal change of coordinates. We give two proofs of this nonvanishing result. The first uses a local perturbation argument. The second reduces it to the absence of a common zero for five explicit polynomials on [−1,1]⁶, verified by a computer-assisted Bernstein subdivision argument.",
      "summary": "A smooth positive density in four dimensions that admits no equipartition into sixteen equal parts by four affine hyperplanes.",
      "methods": [],
      "related_problems": [],
      "resources": [
        {
          "label": "Verification files on arXiv",
          "url": "https://arxiv.org/abs/2608.23312#anc"
        }
      ],
      "videos": [],
      "date": "2026-08-24"
    },
    {
      "id": "moment-curve-blocking",
      "title": "Blocking codimension-one simplices on the moment curve",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "convexity"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/moment-curve-blocking/",
      "arxiv": "2608.09829",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "selection-borsuk-ulam",
      "title": "Selection-structure generalizations of the Borsuk–Ulam theorem",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "preprint",
      "type": "research",
      "reference": "",
      "topics": [
        "topological-combinatorics",
        "tverberg"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/selection-borsuk-ulam/",
      "arxiv": "2607.21410",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "beyond-matroids",
      "title": "KKM theorems and discrete geometry beyond matroids",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "topological-combinatorics",
        "helly",
        "tverberg"
      ],
      "student_authors": [],
      "featured": true,
      "url": "https://www.psoberon.com/papers/beyond-matroids/",
      "arxiv": "2606.16832",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "Selection structures extend the role of color classes and matroids in KKM theorems and discrete geometry.",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": [],
      "date": "2026-06-15"
    },
    {
      "id": "convex-splinters",
      "title": "Variations of Helly's theorem for convex splinters",
      "authors": [
        "Pablo Soberón",
        "Iris Ye"
      ],
      "year": 2026,
      "status": "preprint",
      "type": "research",
      "reference": "",
      "topics": [
        "helly",
        "tverberg",
        "convexity"
      ],
      "student_authors": [
        "Iris Ye"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/convex-splinters/",
      "arxiv": "2606.23437",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "tverberg-cores",
      "title": "Tverberg cores and Kalai’s cascade conjecture",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "tverberg",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": true,
      "url": "https://www.psoberon.com/papers/tverberg-cores/",
      "arxiv": "2605.21305",
      "doi": null,
      "pdf": null,
      "abstract": "We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an n-simplex to ℝᵈ, let Tᵣ(f) be the set of points contained in the images of r pairwise disjoint faces. We prove that if r is a prime power and dim Tᵣ(f) ≤ k, then there exists a point that remains an r-Tverberg point after any t vertices are deleted, provided n = (r−1)(d+1) + t(k+1).\n\nFor t=1, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is 0-dimensional.",
      "summary": "Topological Tverberg cores, vertex-deletion tolerance, and consequences for Kalai’s cascade conjecture.",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": [],
      "date": "2026-05-20"
    },
    {
      "id": "sierksma-seven-points",
      "title": "An elementary proof of Sierksma's conjecture for seven points in the plane",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "tverberg"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/sierksma-seven-points/",
      "arxiv": "2604.18485",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "high-dimensional-envy-free",
      "title": "High-dimensional envy-free partitions",
      "authors": [
        "Christina Yu",
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "published",
      "type": "research",
      "reference": "Comb. Theory 6, no. 1, Paper No. 11, 19 pp.",
      "topics": [
        "fair-division",
        "topological-combinatorics"
      ],
      "student_authors": [
        "Christina Yu"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/high-dimensional-envy-free/",
      "arxiv": "2311.09905",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "fair-distribution-bundles",
      "title": "Fair distribution of bundles",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2026,
      "status": "published",
      "type": "research",
      "reference": "Proc. Amer. Math. Soc. 154, no. 7, 3085–3097.",
      "topics": [
        "fair-division",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/fair-distribution-bundles/",
      "arxiv": "2507.13421",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "mass-assignments-spheres-wedges",
      "title": "Partitions of mass assignments with spheres and wedges",
      "authors": [
        "Deron Lessure",
        "Pablo Soberón"
      ],
      "year": 2025,
      "status": "accepted",
      "type": "research",
      "reference": "Accepted in European Journal of Combinatorics",
      "topics": [
        "mass-partitions"
      ],
      "student_authors": [
        "Deron Lessure"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/mass-assignments-spheres-wedges/",
      "arxiv": "2507.06919",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "krasnosselsky-colorful-quantitative",
      "title": "Colorful and Quantitative Variations of Krasnosselsky's Theorem",
      "authors": [
        "Connor Donovan",
        "Danielle Paulson",
        "Pablo Soberón"
      ],
      "year": 2025,
      "status": "published",
      "type": "research",
      "reference": "Studia Sci. Math. Hungar. 62 (2025), no. 4, 336–353.",
      "topics": [
        "helly",
        "convexity"
      ],
      "student_authors": [
        "Connor Donovan",
        "Danielle Paulson"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/krasnosselsky-colorful-quantitative/",
      "arxiv": "2304.04828",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "tverberg-partition-graphs",
      "title": "Tverberg Partition Graphs",
      "authors": [
        "Deborah Oliveros",
        "Érika Roldán",
        "Pablo Soberón",
        "Antonio J. Torres"
      ],
      "year": 2025,
      "status": "published",
      "type": "research",
      "reference": "SIAM J. Discrete Math. 39 (2), 863–880.",
      "topics": [
        "tverberg",
        "graph-theory"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/tverberg-partition-graphs/",
      "arxiv": "2310.08563",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "discrete-helly-boxes",
      "title": "Extensions of discrete Helly theorems for boxes",
      "authors": [
        "Timothy Edwards",
        "Pablo Soberón"
      ],
      "year": 2025,
      "status": "published",
      "type": "research",
      "reference": "SIAM J. Discrete Math. 39 (2), 1349–1362.",
      "topics": [
        "helly",
        "convexity"
      ],
      "student_authors": [
        "Timothy Edwards"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/discrete-helly-boxes/",
      "arxiv": "2404.14308",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "tverberg-support-vector-machines",
      "title": "Tverberg's theorem and multi-class support vector machines",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2025,
      "status": "published",
      "type": "research",
      "reference": "Found. Data Sci. 7 (2), 568–580.",
      "topics": [
        "tverberg"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/tverberg-support-vector-machines/",
      "arxiv": "2404.16724",
      "doi": "10.3934/fods.2024039",
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "cookie-cutters",
      "title": "Cookie cutters: Bisections with fixed shapes",
      "authors": [
        "Patrick Schnider",
        "Pablo Soberón"
      ],
      "year": 2025,
      "status": "published",
      "type": "research",
      "reference": "Adv. in Appl. Math. 171, Paper No. 102957, 15 pp.",
      "topics": [
        "mass-partitions"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/cookie-cutters/",
      "arxiv": "2502.17176",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "yao-yao-central-transversals",
      "title": "Generalizations of the Yao–Yao partition theorem and the central transversal theorem",
      "authors": [
        "Michael N. Manta",
        "Pablo Soberón"
      ],
      "year": 2024,
      "status": "published",
      "type": "research",
      "reference": "Discrete Comput. Geom. 71 (4), 1381–1402.",
      "topics": [
        "mass-partitions"
      ],
      "student_authors": [
        "Michael N. Manta"
      ],
      "featured": true,
      "url": "https://www.psoberon.com/papers/yao-yao-central-transversals/",
      "arxiv": "2107.06233",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "mass-assignments-flags",
      "title": "Bisections of mass assignments using flags of affine spaces",
      "authors": [
        "Ilani Axelrod-Freed",
        "Pablo Soberón"
      ],
      "year": 2024,
      "status": "published",
      "type": "research",
      "reference": "Discrete Comput. Geom. 72 (2), 550–568.",
      "topics": [
        "mass-partitions",
        "topological-combinatorics"
      ],
      "student_authors": [
        "Ilani Axelrod-Freed"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/mass-assignments-flags/",
      "arxiv": "2109.13106",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "tverberg-polytopes",
      "title": "Improved Tverberg theorems for certain families of polytopes",
      "authors": [
        "Pablo Soberón",
        "Shira Zerbib"
      ],
      "year": 2024,
      "status": "accepted",
      "type": "research",
      "reference": "Accepted to Discrete Comput. Geom.",
      "topics": [
        "tverberg"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/tverberg-polytopes/",
      "arxiv": "2404.11533",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "mass-assignments-parallel",
      "title": "Bisections of mass assignments by parallel hyperplanes",
      "authors": [
        "Nikola Sadovek",
        "Pablo Soberón"
      ],
      "year": 2024,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "mass-partitions",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/mass-assignments-parallel/",
      "arxiv": "2412.04058",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "complex-tverberg-vrecica",
      "title": "Complex analogues of the Tverberg–Vrećica conjecture and central transversal theorems",
      "authors": [
        "Nikola Sadovek",
        "Pablo Soberón"
      ],
      "year": 2024,
      "status": "published",
      "type": "research",
      "reference": "Trans. Amer. Math. Soc. 379 (2026), no. 5, 3665–3691.",
      "topics": [
        "mass-partitions",
        "tverberg",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": true,
      "url": "https://www.psoberon.com/papers/complex-tverberg-vrecica/",
      "arxiv": "2408.14337",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
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      "videos": [],
      "publication_year": 2026
    },
    {
      "id": "bisecting-parallel-families",
      "title": "Bisecting masses with families of parallel hyperplanes",
      "authors": [
        "Alfredo Hubard",
        "Pablo Soberón"
      ],
      "year": 2024,
      "status": "submitted",
      "type": "research",
      "reference": "",
      "topics": [
        "mass-partitions"
      ],
      "student_authors": [],
      "featured": true,
      "url": "https://www.psoberon.com/papers/bisecting-parallel-families/",
      "arxiv": "2404.14320",
      "doi": null,
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
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      "videos": []
    },
    {
      "id": "hyperplane-depth",
      "title": "Combinatorial Depth Measures for Hyperplane Arrangements",
      "authors": [
        "Patrick Schnider",
        "Pablo Soberón"
      ],
      "year": 2023,
      "status": "published",
      "type": "research",
      "reference": "39th International Symposium on Computational Geometry, Art. No. 55, 14 pp., LIPIcs 258.",
      "topics": [
        "tverberg",
        "convexity"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/hyperplane-depth/",
      "arxiv": "2302.07768",
      "doi": "10.4230/LIPIcs.SoCG.2023.55",
      "pdf": null,
      "abstract": "",
      "summary": "",
      "methods": [],
      "related_problems": [],
      "resources": [],
      "videos": []
    },
    {
      "id": "lifting-mass-partitions",
      "title": "Lifting methods in mass partition problems",
      "authors": [
        "Pablo Soberón",
        "Yuki Takahashi"
      ],
      "year": 2023,
      "status": "published",
      "type": "research",
      "reference": "Int. Math. Res. Not. IMRN 2023 (16), 14103–14130.",
      "topics": [
        "mass-partitions",
        "topological-combinatorics"
      ],
      "student_authors": [
        "Yuki Takahashi"
      ],
      "featured": false,
      "url": "https://www.psoberon.com/papers/lifting-mass-partitions/",
      "arxiv": "2109.03749",
      "doi": "10.1093/imrn/rnac224",
      "pdf": null,
      "abstract": "",
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      "methods": [],
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      "videos": []
    },
    {
      "id": "more-participants-than-allocations",
      "title": "Fair distributions for more participants than allocations",
      "authors": [
        "Pablo Soberón"
      ],
      "year": 2022,
      "status": "published",
      "type": "research",
      "reference": "Proc. Amer. Math. Soc., Ser. B 9, 404–414.",
      "topics": [
        "fair-division",
        "topological-combinatorics"
      ],
      "student_authors": [],
      "featured": false,
      "url": "https://www.psoberon.com/papers/more-participants-than-allocations/",
      "arxiv": "2110.03600",
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