International Journal of Data Science in the Mathematical Sciences

Calabi–Yau links are specific $S^1$-fibrations over Calabi–Yau manifolds, when the link is 7-dimensional they exhibit both Sasakian and G2 structures. In this invited contribution to the DANGER proceedings, previous work exhaustively computing Calabi–Yau links and selected topological properties is summarized. Machine learning of these properties inspires new conjectures about their computation, as well as the respective Gröbner bases.

In this paper, we report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers. We show experimentally that these models achieve high accuracy by learning a combination of Mestre–Nagao-type heuristics and murmurations, noting that the relative importance of these features varies with degree. The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time. We give experimental evidence and provide heuristics that suggest this cannot be done with standard machine learning techniques.

From $n$-dimensional reflexive polytopes one can construct $n-1$ complex-dimensional Calabi–Yau manifolds as hypersurfaces in toric varieties. In this contribution to the DANGER 3: Data Numbers and Geometry conference proceedings, we summarise previous work [P. Berglund, Y.-H. He, E. Heyes, E. Hirst, V. Jejjala and A. Lukas, New Calabi–Yau Manifolds from Genetic Algorithms (2023)] generating reflexive polytopes using genetic algorithms. As a proof of principle, we demonstrate that the genetic algorithm can reproduce known reflexive polytopes in two, three and four dimensions. Motivated by this result, we construct five-dimensional reflexive polytopes and establish that many of these are not in existing datasets and therefore give rise to new Calabi–Yau four-folds.

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre–Nagao sum

Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves [Y.-H. He, K.-H. Lee, T. Oliver and A. Pozdnyakov, Murmurations of elliptic curves, preprint (2022), arXiv:2204.10140.]. Surprisingly, this suggests that in some cases, opting for a smaller value of $B$ yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of $[40000,45000]$, the rank classification based on $a_p$’s with $p<B=3200$ produces better results compared to using $B=50000$. This phenomenon finds partial explanation in the recent work of Zubrilina [Murmurations (2023)].

Parameterized systems of polynomial equations arise in many applications including computer vision, chemistry, and kinematics. Numerical homotopy continuation methods are a fundamental technique within numerical algebraic geometry for both solving these polynomial systems and determining more refined information about their structure. Imperative to these solving methods is the use of data either synthetic or from the application itself, such as image pixel data for computer vision and leg length parameters for kinematics. This paper will describe projects that incorporate the use of data to find real solutions and/or the structure of real solutions for problems in applications. Illustrative examples are given to highlight various uses of data within computer vision, machine learning, and kinematics applications.

The Kronecker coefficients are the decomposition multiplicities of the tensor product of two irreducible representations of the symmetric group. Unlike the Littlewood–Richardson coefficients, which are the analogues for the general linear group, there is no known combinatorial description of the Kronecker coefficients, and it is an NP-hard problem to decide whether a given Kronecker coefficient is zero or not. In this paper, we show that standard machine learning algorithms such as Nearest Neighbors, Convolutional Neural Networks and Gradient Boosting Decision Trees may be trained to predict whether a given Kronecker coefficient is zero or not. Our results show that a trained machine can efficiently perform this binary classification with high accuracy ($\approx$0.98).