Machine learning detects terminal singularities

• URL: http://arxiv.org/abs/2310.20458v1
• Date: Tue, 31 Oct 2023 13:51:24 GMT
• Title: Machine learning detects terminal singularities
• Authors: Tom Coates, Alexander M. Kasprzyk, Sara Veneziale
• Abstract summary: Q-Fano varieties are positively curved shapes which have Q-factorial terminal singularities. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification.
• Score: 49.1574468325115
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are "atomic pieces" of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on 8-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank 2, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fanos in dimension 8. How the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future. Inspired by the ML analysis, we formulate and prove a new global combinatorial criterion for a positively curved toric variety of Picard rank 2 to have terminal singularities. Together with the first sketch of the landscape of Q-Fanos in higher dimensions, this gives new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.

Related papers

• Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes [4.229995708813431]
  We use graph neural networks and AI explainability techniques to discover mutation equivalence criteria for quivers of type $tildeD_n$. We also show that our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D_n$.
  arXiv  Detail & Related papers  (2024-11-12T01:09:41Z)
• A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems [87.30652640973317]
  Recent advances in computational modelling of atomic systems represent them as geometric graphs with atoms embedded as nodes in 3D Euclidean space. Geometric Graph Neural Networks have emerged as the preferred machine learning architecture powering applications ranging from protein structure prediction to molecular simulations and material generation. This paper provides a comprehensive and self-contained overview of the field of Geometric GNNs for 3D atomic systems.
  arXiv  Detail & Related papers  (2023-12-12T18:44:19Z)
• Machine learning the dimension of a Fano variety [49.1574468325115]
  We show that a simple feed-forward neural network can determine the dimension of X. We also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.
  arXiv  Detail & Related papers  (2023-09-11T14:13:30Z)
• Dynamics and Geometry of Entanglement in Many-Body Quantum Systems [0.0]
  A new framework is formulated to study entanglement dynamics in many-body quantum systems. The Quantum Correlation Transfer Function (QCTF) is transformed into a new space of complex functions with isolated singularities. The QCTF-based geometric description offers the prospect of theoretically revealing aspects of many-body entanglement.
  arXiv  Detail & Related papers  (2023-08-18T19:16:44Z)
• QH9: A Quantum Hamiltonian Prediction Benchmark for QM9 Molecules [69.25826391912368]
  We generate a new Quantum Hamiltonian dataset, named as QH9, to provide precise Hamiltonian matrices for 999 or 2998 molecular dynamics trajectories. We show that current machine learning models have the capacity to predict Hamiltonian matrices for arbitrary molecules.
  arXiv  Detail & Related papers  (2023-06-15T23:39:07Z)
• A Unified Algebraic Perspective on Lipschitz Neural Networks [88.14073994459586]
  This paper introduces a novel perspective unifying various types of 1-Lipschitz neural networks. We show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers.
  arXiv  Detail & Related papers  (2023-03-06T14:31:09Z)
• A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes [78.120734120667]
  We build the preimage of a point in the output manifold in the input space. We focus for simplicity on the case of neural networks maps from n-dimensional real spaces to (n - 1)-dimensional real spaces.
  arXiv  Detail & Related papers  (2021-12-17T11:47:45Z)
• Geometry of quantum hydrodynamics in theoretical chemistry [0.0]
  This thesis investigates geometric approaches to quantum hydrodynamics (QHD) in order to develop applications in theoretical quantum chemistry. The momentum map approach to QHD is then applied to the nuclear dynamics in a chemistry model known as exact factorization. A new mixed quantum-classical model is then derived by considering a generalised factorisation ansatz.
  arXiv  Detail & Related papers  (2020-09-28T19:58:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.