We apply a variety of machine learning methods to the study of Seiberg duality within 4d $\mathcal{N}=1$ quantum field theories arising on the worldvolumes of D3-branes probing toric Calabi-Yau 3-folds. Such theories admit an elegant description in terms of bipartite tessellations of the torus known as brane tilings or dimer models. An intricate network of infrared dualities interconnects the space of such theories and partitions it into universality classes, the prediction and classification of which is a problem that naturally lends itself to a machine learning investigation. In this paper, we address a preliminary set of such enquiries. We begin by training a fully connected neural network to identify classes of Seiberg dual theories realised on $\mathbb{Z}_m\times\mathbb{Z}_n$ orbifolds of the conifold and achieve $R^2=0.988$. Then, we evaluate various notions of robustness of our methods against perturbations of the space of theories under investigation, and discuss these results in terms of the nature of the neural network's learning. Finally, we employ a more sophisticated residual architecture to classify the toric phase space of the $Y^{6,0}$ theories, and to predict the individual gauged linear $\sigma$-model multiplicities in toric diagrams thereof. In spite of the non-trivial nature of this task, we achieve remarkably accurate results; namely, upon fixing a choice of Kasteleyn matrix representative, the regressor achieves a mean absolute error of $0.021$. We also discuss how the performance is affected by relaxing these assumptions.

We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N=1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N=1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy.

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past year, this writing is a very preliminary draft of a book to appear with Springer, by whose kind permission we post to ArXiv for comments and suggestions.