Details
| Original language | English |
|---|---|
| Article number | 24 |
| Journal | Advances in Applied Clifford Algebras |
| Volume | 32 |
| Issue number | 2 |
| Early online date | 22 Feb 2022 |
| Publication status | Published - Apr 2022 |
Abstract
The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of 96 % of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices.
Keywords
- Artificial Intelligence, K3 Surface, Neural Network, Numerical and Symbolic Computation, Period, Picard Group
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
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In: Advances in Applied Clifford Algebras, Vol. 32, No. 2, 24, 04.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Deep Learning Gauss–Manin Connections
AU - Heal, Kathryn
AU - Kulkarni, Avinash
AU - Sertöz, Emre Can
N1 - Funding Information: All three authors were partially funded by Max Planck Institute for Mathematics in the Sciences (MPI MiS) Leipzig. In addition to that: K. Heal was funded by Harvard University and C. S. Draper Laboratory. A. Kulkarni was funded by Dartmouth College and the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation (Simons Foundation grant 550033). E.C. Sertöz was funded by Leibniz University Hannover. All of the educational institutions provided computational resources.
PY - 2022/4
Y1 - 2022/4
N2 - The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of 96 % of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices.
AB - The Gauss–Manin connection of a family of hypersurfaces governs the change of the period matrix along the family. This connection can be complicated even when the equations defining the family look simple. When this is the case, it is expensive to compute the period matrices of varieties in the family via homotopy continuation. We train neural networks that can quickly and reliably guess the complexity of the Gauss–Manin connection of pencils of hypersurfaces. As an application, we compute the periods of 96 % of smooth quartic surfaces in projective 3-space whose defining equation is a sum of five monomials; from the periods of these quartic surfaces, we extract their Picard lattices and the endomorphism fields of their transcendental lattices.
KW - Artificial Intelligence
KW - K3 Surface
KW - Neural Network
KW - Numerical and Symbolic Computation
KW - Period
KW - Picard Group
UR - http://www.scopus.com/inward/record.url?scp=85125394686&partnerID=8YFLogxK
U2 - 10.1007/s00006-022-01207-1
DO - 10.1007/s00006-022-01207-1
M3 - Article
AN - SCOPUS:85125394686
VL - 32
JO - Advances in Applied Clifford Algebras
JF - Advances in Applied Clifford Algebras
SN - 0188-7009
IS - 2
M1 - 24
ER -