Learning Fricke signs from Maass form Coefficients

In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered "murmuration" phenomenon, and that these patterns become more pronounced when parity is incorporated as an additional feature. Approximately 43% of the forms in our dataset have an unknown Fricke sign. For the remaining forms, we employ Linear Discriminant Analysis (LDA) to machine learn their Fricke sign, achieving 96% (resp. 94%) accuracy for forms with even (resp. odd) parity. We apply the trained LDA model to forms with unknown Fricke signs to make predictions. The average values based on the predicted Fricke signs are computed and compared to those for forms with known signs to verify the reasonableness of the predictions. Additionally, a subset of these predictions is evaluated against heuristic guesses provided by Hejhal's algorithm, showing a match approximately 95% of the time. We also use neural networks to obtain results comparable to those from the LDA model.

Neural Vector Tomography for Reconstructing a Magnetization Vector Field

Discretized techniques for vector tomographic reconstructions are prone to producing artifacts in the reconstructions. The quality of these reconstructions may further deteriorate as the amount of noise increases. In this work, we instead model the underlying vector fields using smooth neural fields. Owing to the fact that the activation functions in the neural network may be chosen to be smooth and the domain is no longer pixelated, the model results in high-quality reconstructions, even under presence of noise. In the case where we have underlying global continuous symmetry, we find that the neural network substantially improves the accuracy of the reconstruction over the existing techniques.

cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature

We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

Machine Learned Calabi--Yau Metrics and Curvature

Finding Ricci-flat (Calabi--Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi--Yau metric within a given K\"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi--Yau threefolds. Using these Ricci-flat metric approximations for the Cefal\'u and Dwork family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points, but also elsewhere. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations.