Research

Forschungsschwerpunkte/Research Interests

 

  • Konvexe, diskrete und torische Geometrie, mit einem Schwerpunkt auf Gitterpolytopen
  • Convex, discrete and toric geometry with a focus on lattice polytopes

Publikationen/Publications

    1. Tautological systems, homogeneous spaces and the holonomic rank problem, [arXiv], Paul Görlach, Thomas Reichelt, Christian Sevenheck, Avi Steiner, Uli Walther, November 2022.
    2. Thin polytopes: Lattice polytopes with vanishing local h∗-polynomial, [arXiv], Christopher Borger, Andreas Kretschmer, Benjamin Nill, July 2022.
    3. Proof of a conjecture of Batyrev and Juny on Gorenstein polytopes, [arXiv], Benjamin Nill, March 2022. 
    4. Sharp bounds on fake weighted projective spaces with canonical singularities, [arXiv], Gennadiy Averkov, Alexander Kasprzyk, Martin Lehmann, Benjamin Nill, May 2021. 
    5. Empty simplices of large width, [arXiv], Joseph Doolittle, Lukas Katthän, Benjamin Nill, Francisco Santos, March 2021.

    6. Polynomial systems arising from Nash equilibria - A story about how game theory meets multilinear systems and product of simplices, Julia HomotopyContinuation.jl, Irem Portakal.

    7. Rigid Gorenstein toric Fano varieties arising from directed graphs, [arXiv], Selvi Kara, Irem Portakal, and Akiyoshi Tsuchiya March 2021, submitted.

    8. Rigid toric matrix Schubert varieties, [arXiv], Irem Portakal, January 2020, submitted.

    9. Generalized flatness constants, spanning lattice polytopes, and the Gromov width, [arXiv], Gennadiy Averkov, Johannes Hofscheier, Benjamin Nill.

    10. On rigidity of toric varieties arising from bipartite graphs, [arXiv:1905.02445], Journal of Algebra, Volume 569, 2021, Pages 784-822, ISSN 0021-8693, https://doi.org/10.1016/j.jalgebra.2020.09.032.,[arXiv], Irem Portakal,

    11. A note on deformations and mutations of fake weighted projective planes, Irem Portakal, Algebraic and Geometric Combinatorics on Lattice Polytopes, Proceedings of the 2018 Summer Workshop on Lattice Polytopes, June 2019. [arXiv]

    12. Families of lattice polytopes of mixed degree one, [arXiv], Gabriele Balletti, Christopher Borger, accepted in Journal of Combinatorial Theory, Series A.

    13. Classification of triples of lattice polytopes with a given mixed volume, [arXiv], Gennadiy Averkov, Christopher Borger, Ivan Soprunov, accepted in Discrete & Computational Geometry.

    14. Stability of tangent bundles on smooth toric Picard-rank-2 varieties and surfaces, [arXiv], Milena Hering, Benjamin Nill, Hendrik Süß.

    15. On defectivity of families of full-dimensional point configurations, [arXiv], Christopher Borger, Benjamin Nill, submitted to Proceedings of the AMS.

    16. Spanning Lattice Polytopes and the Uniform Position Principle [arxiv], Johannes Hofscheier, Lukas Katthän, Benjamin Nill.

    17. Lattice simplices with a fixed positive number of interior lattice points: A nearly optimal volume bound [arxiv], Gennadiy Averkov, Jan Krümpelmann, Benjamin Nill.

    18. The mixed degree of families of lattice polytopes [arxiv], Benjamin Nill.

    19. Smooth polytopes with negative Ehrhart coefficients [arxiv], Federico Castillo, Fu Liu, Benjamin Nill, Andreas Paffenholz, Journal of Combinatorial Theory Ser. A.

    20. On the maximum dual volume of a canonical Fano polytope [arxiv], Gabriele Balletti, Alexander M. Kasprzyk, Benjamin Nill.

    21. On Ehrhart polynomials of lattice triangles [arxiv], Johannes Hofscheier, Benjamin Nill, Dennis Öberg, Electron. J. Combin. 25 (2018), no. 1, Paper 1.3, 8 pp.

    22. Discrete Mixed Volume and Hodge-Deligne Numbers [arxiv], Sandra Di Rocco, Christian Haase, Benjamin Nill.

    23. Ehrhart Theory of Spanning Lattice Polytopes [arxiv], Johannes Hofscheier, Lukas Katthän, Benjamin Nill; Int. Math. Res. Not. IMRN 2018, no. 19, 5947-5973.

    24. Gorenstein polytopes with trinomial h*-polynomials [arxiv], Benjamin Nill, Akihiro Higashitani, Akiyoshi Tsuchiya; March 2015.

    25. Minimality and mutation-equivalence of polygons [arxiv], Benjamin Nill, Alexander Kasprzyk, Thomas Prince; January 2015.

    26. Toric Fano manifolds with large Picard number [arxiv] Benjamin Nill, Benjamin Assarf; September 2014.

    27. On smooth Gorenstein polytopes [arxiv, database] Benjamin Nill, Benjamin Lorenz; March 2013. Accepted in: Tohoku Math. J.

    28. The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes [arxiv]. Benjamin Nill, Arnau Padrol; September 2012.
      Accepted in: European Journal of Combinatorics.

    29. Few smooth d-polytopes with N lattice points [arxiv] Benjamin Nill, Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Andreas Paffenholz, Günter Rote, Francisco Santos, Hal Schenck; October 2010. Israel Journal of Mathematics, 207: 301-329, 2015.

    30. Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results [arxiv, article], Benjamin Nill, Gennadiy Averkov, Jan Krümpelmann. Advances in Mathematics, 274: 118-166, 2015.

    31. Polytopes associated to Dihedral Groups [arxiv, article], Benjamin Nill, Barbara Baumeister, Christian Haase, Andreas Paffenholz.
      Ars Mathematica Contemporanea, 7(1), 2014.

    32. Fano polytopes [edoc,book], Benjamin Nill, Alexander Kasprzyk.In: Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer; edited by: Anton Rebhan, Ludmil Katzarkov, Johanna Knapp, Radoslav Rashkov, Emanuel Scheidegger; World Scientific, 2012.

    33. On the equality case in Ehrhart's volume conjecture [arxiv, article] Benjamin Nill, Andreas Paffenholz. Advances in Geometry, 14: 579–586, 2014

    34. A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes [arxiv, article], Benjamin Nill, Alicia Dickenstein, Michèle Vergne. C. R. Math. Acad. Sci. Paris, 350: 229-233, 2012.

    35. Reflexive polytopes of higher index and the number 12 [arxiv, article] Benjamin Nill, Alexander Kasprzyk. Electronic Journal of Combinatorics, 19: #P9, 2012.

    36. Polyhedral adjunction theory [arxiv, article], Benjamin Nill, Sandra Di Rocco, Christian Haase, Andreas Paffenholz. Algebra & Number Theory, 7(10): 2417-2446, 2013.

    37. Projecting lattice polytopes without interior lattice points [arxiv, article] Benjamin Nill, Günter Ziegler. Mathematics of Operations Research, 36: 462-467, 2011.

    38. Gorenstein polytopes and their stringy E-functions [arxiv, article], Benjamin Nill, Jan Schepers. Mathematische Annalen, 355: 457-480, 2013.

    39. A simple combinatorial criterion for projective toric manifolds with dual defect [arxiv, article], Benjamin Nill, Alicia Dickenstein. Mathematical Research Letters, 17: 435-448, 2010.

    40. Examples of non-symmetric Kähler-Einstein toric Fano manifolds [arxiv, article], Benjamin Nill, Andreas Paffenholz. Beiträge zur Algebra und Geometrie, 52: 279-304, 2011.

    41. Lattice width directions and Minkowski's 3^d-theorem [arxiv, article], Benjamin Nill, Jan Draisma, Tyrrell B. McAllister. SIAM Journal on Discrete Mathematics, 26: 1104-1107, 2012.

    42. On the combinatorial classification of toric log Del Pezzo surfaces [arxiv, article, www], Benjamin Nill, Alexander M. Kasprzyk, Maximilian Kreuzer.
      LMS Journal of Computation and Mathematics, 13: 33-46, 2010.

    43. Q-factorial Gorenstein toric Fano varieties with large Picard number [arxiv, article], Benjamin Nill, Mikkel Øbro. Tohoku Mathematical Journal, 62: 1-15, 2010.

    44. Stanley's conjecture, cover depth and extremal simplicial complexes [article], Benjamin Nill, Kathrin Vorwerk. Le Matematiche, 63: 213-228, 2008.

    45. Flow polytopes and the graph of reflexive polytopes [edoc, article], Benjamin Nill, Klaus Altmann, Sabine Schwentner, Izolda Wiercinska. Discrete Mathematics, 309: 4992-4999, 2009.

    46. Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials [arxiv, article], Benjamin Nill, Christian Haase, Sam Payne. Journal für die reine und angewandte Mathematik, 637: 207-216, 2009

    47. Lattice points in Minkowski sums [arxiv, article], Benjamin Nill, Christian Haase, Andreas Paffenholz, Francisco Santos. Electronic Journal of Combinatorics, 15: #N11, 2008.

    48. On permutation polytopes [arxiv, article, www], Benjamin Nill, Barbara Baumeister, Christian Haase, Andreas Paffenholz. Advances in Mathematics, 222: 431-452, 2009.

    49. A boundedness result for toric log Del Pezzo surfaces [arxiv, article], Benjamin Nill, Dimitrios I. Dais. Archiv der Mathematik, 91: 526-535, 2008.

    50. Lattice polytopes having h*-polynomials with given degree and linear coefficient [arxiv, article],Benjamin Nill, European Journal of Combinatorics, 29: 1596-1602, 2008.

    51. Combinatorial aspects of mirror symmetry [arxiv, book], Benjamin Nill, Victor Batyrev. Contemporary Mathematics - Integer Points in Polyhedra: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, June 2006, Snowbird, Utah, 452: 35-66, 2008.

    52. Classification of toric Fano 5-folds [arxiv, article, www],Benjamin Nill, Maximilian Kreuzer. Advances in Geometry, 9: 85-97, 2009.

    53. What is the maximal number of vertices of a reflexive polytope? [edoc, book] part of a note entitled 'Let me tell you my favorite lattice point problem ...', Benjamin Nill, Matthias Beck, Bruce Reznick, Carla Savage, Ivan Soprunov, Zhiqiang Xu. Contemporary Mathematics - Integer Points in Polyhedra: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, June 2006, Snowbird, Utah, 452: 179-187, 2008.

    54. Multiples of lattice polytopes without interior lattice points [arxiv, article], Benjamin Nill, Victor Batyrev. Moscow Mathematical Journal, 7: 195-207, 2007.

    55. Classification of pseudo-symmetric simplicial reflexive polytopes [arxiv, book], Benjamin Nill, Contemporary Mathematics - Algebraic and Geometric Combinatorics:
      Proceedings of a Euroconference in Mathematics, August 20-26, 2005, Anogia, Crete, Greece, 423: 269-282, 2006.

    56. Lattices generated by skeletons of reflexive polytopes [arxiv, article], Benjamin Nill, Christian Haase. Journal of Combinatorial Theory, Series A, 115: 340-344, 2008.

    57. Volume and lattice points of reflexive simplices [arxiv, article], Benjamin Nill, Discrete and Computational Geometry, 37(2): 301-320, 2007.

    58. Complete toric varieties with reductive automorphism group [arxiv, article], Benjamin Nill. Mathematische Zeitschrift, 252(4): 767-786, 2006.

    59. Gorenstein toric Fano varieties [arxiv, article], Benjamin Nill, Manuscripta mathematica, 116(2): 183-210, 2005.

 

Other Publications

      1. Rigid toric bipartite graphs, [GitHub]. Irem Portakal, Polymake and Singular scripts for investigating the rigidity of toric varieties arising from a bipartite graph.

      2. Rigidity of toric varieties associated to bipartite graphs, Irem Portakal, PHD Thesis, [online] at Freie Universität Berlin.

      3. Generalized compactifications of Batyrev hypersurface families [arxiv], Benjamin Nill, Karl Fredrickson; October 2014 (intermediate version; final version published by Fredrickson).

      4. Permutation polytopes of cyclic groups [arxiv, edoc], Benjamin Nill, contributed paper (extended abstract of poster) for FPSAC, 2012.

      5. Combinatorial questions related to stringy E-polynomials of Gorenstein polytopes [article], Benjamin Nill, in: Oberwolfach Reports, 9(2): 62-64, 2012.

      6. Gorenstein toric Fano varieties [edoc], Benjamin Nill, Dissertation, July 2005.

 

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