This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.

They can be realized via simple operations from within the group itself, hence the adjective "inner".

The vastness of the string landscape presents a serious computational challenge. This immensity stems from the multitude of choices for the internal manifolds on which string theory is compactified (or for non-geometric constructions, choices of conformal field theory). Even with a fixed compactification manifold, additional discrete choices--such as bundle or brane configurations, and the quantized fluxes threaded through internal cycles--further enlarge the space of solutions. Despite its vastness, the string landscape is conjectured to be finite, in the sense that there are only finitely many low energy effective field theories with a fixed, finite energy cutoff that are consistent with quantum gravity [1-3]. The finiteness of the landscape is both an important premise in the program of landscape statistics [1] and argued to be a universal property of quantum gravity [2]. It is however only when we restrict to very small regions of the landscape, e.g., intersecting D-brane models in a specific Calabi-Yau orientifold, that an exact number of solutions is known [4] (though it was shown earlier that the number is finite [5]). Compactifications of string theory on Calabi-Yau manifolds stand out as an especially well-motivated class of solutions for data mining the landscape. In particular, Calabi-Yau threefolds yield four-dimensional vacuum configurations of superstring theory that can potentially accommodate realistic particle physics coupled to gravity [6].

This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.

Given a continuous finitely piecewise linear function $f:\mathbb{R}^{n_0} \to \mathbb{R}$ and a fixed architecture $(n_0,\ldots,n_k;1)$ of feedforward ReLU neural networks, the exact function realization problem is to determine when some network with the given architecture realizes $f$. To develop a systematic way to answer these questions, we establish a connection between toric geometry and ReLU neural networks. This approach enables us to utilize numerous structures and tools from algebraic geometry to study ReLU neural networks. Starting with an unbiased ReLU neural network with rational weights, we define the ReLU fan, the ReLU toric variety, and the ReLU Cartier divisor associated with the network. This work also reveals the connection between the tropical geometry and the toric geometry of ReLU neural networks. As an application of the toric geometry framework, we prove a necessary and sufficient criterion of functions realizable by unbiased shallow ReLU neural networks by computing intersection numbers of the ReLU Cartier divisor and torus-invariant curves.

High-quality, large-scale corpora are the cornerstone of building foundation models. In this work, we introduce \textsc{MathPile}, a diverse and high-quality math-centric corpus comprising about 9.5 billion tokens. Throughout its creation, we adhered to the principle of ``\emph{less is more}'', firmly believing in the supremacy of data quality over quantity, even in the pre-training phase. Our meticulous data collection and processing efforts included a complex suite of preprocessing, prefiltering, language identification, cleaning, filtering, and deduplication, ensuring the high quality of our corpus. Furthermore, we performed data contamination detection on downstream benchmark test sets to eliminate duplicates. We hope our \textsc{MathPile} can help to enhance the mathematical reasoning abilities of language models. We plan to open-source different versions of \mathpile with the scripts used for processing, to facilitate future developments in this field.

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.