Built on

K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Getting CICY High , Phys. Lett. B 795
Original
1903
Earlier work this paper cites.
J. Halverson, B. Nelson and F. Ruehle, Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning , JHEP 06
Original
1903
Earlier work this paper cites.
Y.-H. He and S.-J. Lee, Distinguishing elliptic fibrations with AI , Phys. Lett. B 798
Original
1904
Earlier work this paper cites.
A. Ashmore, Y.-H. He and B. A. Ovrut, Machine learning Calabi-Yau metrics , 1910.08605
Original
1910
Earlier work this paper cites.
C. T. C. Wall, Classification problems in differential topology. V , Inventiones mathematicae 1
1966
Earlier work this paper cites.
I. M. Gelfand, A. V. Zelevinsky and M. M. Kapranov, Newton polyhedra of principal a-determinants , Soviet Mathematics. Doklady 40
1989
Earlier work this paper cites.
I. M. Gelfand, A. V. Zelevinsky and M. M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra , Leningrad Math. J. 2
1991
Earlier work this paper cites.
T. Oda and H. S. Park, Linear Gale transforms and Gel’fand-Kapranov-Zelevinskij decompositions , Tohoku Math. J. (2) 43
1991
Earlier work this paper cites.
Birkhäuser Boston, Boston, MA, 1994, 10.1007/978-0-8176-4771-1
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants · 1994
Earlier work this paper cites.
V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties , J. Alg. Geom. 3
1994
Earlier work this paper cites.
World Scientific, Singapore, 1994
T. Hubsch, Calabi-Yau manifolds: A Bestiary for physicists · 1994
Earlier work this paper cites.
P. Berglund, S. H. Katz and A. Klemm, Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties , Nucl. Phys. B 456
1995
Earlier work this paper cites.
C. B. Barber, D. P. Dobkin and H. Huhdanpaa, The quickhull algorithm for convex hulls , ACM Transactions on Mathematical Software 22
1996
Earlier work this paper cites.
J. Halverson and C. Long, Statistical Predictions in String Theory and Deep Generative Models , Fortsch. Phys. 68
Original
2001
Earlier work this paper cites.
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions , Adv. Theor. Math. Phys. 4
2002
Earlier work this paper cites.
R. Deen, Y.-H. He, S.-J. Lee and A. Lukas, Machine Learning String Standard Models , 2003.13339
Original
2003
Earlier work this paper cites.
E. E. Anclin, An upper bound for the number of planar lattice triangulations , Journal of Combinatorial Theory, Series A 103
2003
Earlier work this paper cites.

Similar

math/0211268
V. Kaibel and G. M. Ziegler, Counting lattice triangulations , in Surveys in Combinatorics 2003, number 307 in Lond. Math. Soc. Lect. Note Ser , pp. 277–308, Cambridge University Press, 2003 · 2003
Cited alongside, same era.
T. Bachlechner, B. P. Majumder, H. H. Mao, G. W. Cottrell and J. McAuley, Rezero is all you need: Fast convergence at large depth , 2003.04887
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Cited alongside, same era.
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2004
Cited alongside, same era.
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Original
2006

Then

F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape , JHEP 08
Original
2017
Later among the works it cites.
J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson, Machine Learning in the String Landscape , JHEP 09
Original
2017
Later among the works it cites.
K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Machine Learning CICY Threefolds , Phys. Lett. B 785
Original
2018
Later among the works it cites.
C. Jordan, M. Joswig and L. Kastner, Parallel enumeration of triangulations , Electronic Journal of Combinatorics 25
Original
2018
Later among the works it cites.
P. W. Battaglia, J. B. Hamrick, V. Bapst, A. Sanchez-Gonzalez, V. F. Zambaldi, M. Malinowski et al., Relational inductive biases, deep learning, and graph networks , CoRR (2018) , [ 1806.01261 ]

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2020

Bounding the Kreuzer-Skarke Landscape

Demirtas, Mehmet, McAllister, Liam, Rios-Tascon, Andres

Understand

We study Calabi-Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes.

By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer-Skarke list is bounded above by $\binom{14,111}{494} \approx 10^{928}$.
Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2-face of each polytope in the list.
In this way we prove that the number of topologically inequivalent Calabi-Yau hypersurfaces arising from the Kreuzer-Skarke list is bounded above by $10^{428}$.

Built on

K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Getting CICY High , Phys. Lett. B 795
Original
1903
Earlier work this paper cites.
J. Halverson, B. Nelson and F. Ruehle, Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning , JHEP 06
Original
1903
Earlier work this paper cites.
Y.-H. He and S.-J. Lee, Distinguishing elliptic fibrations with AI , Phys. Lett. B 798
Original
1904
Earlier work this paper cites.
A. Ashmore, Y.-H. He and B. A. Ovrut, Machine learning Calabi-Yau metrics , 1910.08605
Original
1910
Earlier work this paper cites.
C. T. C. Wall, Classification problems in differential topology. V , Inventiones mathematicae 1
1966
Earlier work this paper cites.
I. M. Gelfand, A. V. Zelevinsky and M. M. Kapranov, Newton polyhedra of principal a-determinants , Soviet Mathematics. Doklady 40
1989
Earlier work this paper cites.
I. M. Gelfand, A. V. Zelevinsky and M. M. Kapranov, Discriminants of polynomials in several variables and triangulations of Newton polyhedra , Leningrad Math. J. 2
1991
Earlier work this paper cites.
T. Oda and H. S. Park, Linear Gale transforms and Gel’fand-Kapranov-Zelevinskij decompositions , Tohoku Math. J. (2) 43
1991
Earlier work this paper cites.
Birkhäuser Boston, Boston, MA, 1994, 10.1007/978-0-8176-4771-1
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants · 1994
Earlier work this paper cites.
V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties , J. Alg. Geom. 3
1994
Earlier work this paper cites.
World Scientific, Singapore, 1994
T. Hubsch, Calabi-Yau manifolds: A Bestiary for physicists · 1994
Earlier work this paper cites.
P. Berglund, S. H. Katz and A. Klemm, Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties , Nucl. Phys. B 456
1995
Earlier work this paper cites.
C. B. Barber, D. P. Dobkin and H. Huhdanpaa, The quickhull algorithm for convex hulls , ACM Transactions on Mathematical Software 22
1996
Earlier work this paper cites.
J. Halverson and C. Long, Statistical Predictions in String Theory and Deep Generative Models , Fortsch. Phys. 68
Original
2001
Earlier work this paper cites.
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions , Adv. Theor. Math. Phys. 4
2002
Earlier work this paper cites.
R. Deen, Y.-H. He, S.-J. Lee and A. Lukas, Machine Learning String Standard Models , 2003.13339
Original
2003
Earlier work this paper cites.
E. E. Anclin, An upper bound for the number of planar lattice triangulations , Journal of Combinatorial Theory, Series A 103
2003
Earlier work this paper cites.

Similar

math/0211268
V. Kaibel and G. M. Ziegler, Counting lattice triangulations , in Surveys in Combinatorics 2003, number 307 in Lond. Math. Soc. Lect. Note Ser , pp. 277–308, Cambridge University Press, 2003 · 2003
Cited alongside, same era.
T. Bachlechner, B. P. Majumder, H. H. Mao, G. W. Cottrell and J. McAuley, Rezero is all you need: Fast convergence at large depth , 2003.04887
Original
2003
Cited alongside, same era.
B. Allanach, D. Grellscheid and F. Quevedo, Genetic algorithms and experimental discrimination of SUSY models , JHEP 07
2004
Cited alongside, same era.
M. Kreuzer and H. Skarke, Palp: A package for analysing lattice polytopes with applications to toric geometry , Computer Physics Communications 157
2004
Cited alongside, same era.
Y.-H. He and S.-T. Yau, Graph Laplacians, Riemannian Manifolds and their Machine-Learning , 2006.16619
Original
2006

Then

F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape , JHEP 08
Original
2017
Later among the works it cites.
J. Carifio, J. Halverson, D. Krioukov and B. D. Nelson, Machine Learning in the String Landscape , JHEP 09
Original
2017
Later among the works it cites.
K. Bull, Y.-H. He, V. Jejjala and C. Mishra, Machine Learning CICY Threefolds , Phys. Lett. B 785
Original
2018
Later among the works it cites.
C. Jordan, M. Joswig and L. Kastner, Parallel enumeration of triangulations , Electronic Journal of Combinatorics 25
Original
2018
Later among the works it cites.
P. W. Battaglia, J. B. Hamrick, V. Bapst, A. Sanchez-Gonzalez, V. F. Zambaldi, M. Malinowski et al., Relational inductive biases, deep learning, and graph networks , CoRR (2018) , [ 1806.01261 ]

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

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Cited alongside, same era.

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R. Bagnara, P. M. Hill and E. Zaffanella, The parma polyhedra library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems , Science of Computer Programming 72

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Cited alongside, same era.

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Original

2014

Cited alongside, same era.

R. Altman, J. Gray, Y.-H. He, V. Jejjala and B. D. Nelson, A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List , JHEP 02

Original

2015