arXiv subjects

Edward Hirst

Publications and source records attributed to Edward Hirst.

35 records · Page 2Linked to original sources

arXiv2024

Cluster Algebras: Network Science and Machine Learning

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to classify cluster algebras using the data of seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.

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arXiv2024

Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds

We propose a machine learning approach to study topological quantities related to the Sasakian and $G_2$-geometries of contact Calabi-Yau $7$-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-Nördstrom invariant of the natural $G_2$-structure of the $7$-dimensional link of a weighted projective Calabi-Yau $3$-fold hypersurface singularity, for 7549 of the 7555 possible $\mathbb{P}^4(\textbf{w})$ projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the $\mathbb{P}^4(\textbf{w})$ weights alone, using both neural networks and a symbolic regressor which achieve $R^2$ scores of 0.969 and 0.993 respectively. Additionally, properties of the respective Gröbner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised.

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arXiv2024

Calabi-Yau Links and Machine Learning

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 summarised. Machine learning of these properties inspires new conjectures about their computation, as well as the respective Gröbner bases.

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arXiv2023

Machine Learning in Physics and Geometry

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to help one do mathematics. This is an invited chapter contribution to Elsevier's Handbook of Statistics, Volume 49: Artificial Intelligence edited by S.~G.~Krantz, A.~S.~R.~Srinivasa Rao, and C.~R.~Rao.

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arXiv2022

Mahler Measuring the Genetic Code of Amoebae

Amoebae from tropical geometry and the Mahler measure from number theory play important roles in quiver gauge theories and dimer models. Their dependencies on the coefficients of the Newton polynomial closely resemble each other, and they are connected via the Ronkin function. Genetic symbolic regression methods are employed to extract the numerical relationships between the 2d and 3d amoebae components and the Mahler measure. We find that the volume of the bounded complement of a d-dimensional amoeba is related to the gas phase contribution to the Mahler measure by a degree-d polynomial, with d = 2 and 3. These methods are then further extended to numerical analyses of the non-reflexive Mahler measure. Furthermore, machine learning methods are used to directly learn the topology of 3d amoebae, with strong performance. Additionally, analytic expressions for boundaries of certain amoebae are given.

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arXiv2022

Brain Webs for Brane Webs

We propose a new technique for classifying 5d Superconformal Field Theories arising from brane webs in Type IIB String Theory, using technology from Machine Learning to identify different webs giving rise to the same theory. We concentrate on webs with three external legs, for which the problem is analogous to that of classifying sets of 7-branes. Training a Siamese Neural Network to determine equivalence between any two brane webs shows an improved performance when webs are considered equivalent under a weaker set of conditions. Thus, Machine Learning teaches us that the conjectured classification of 7-brane sets is not complete, which we confirm with explicit examples.

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arXiv2022

Neurons on Amoebae

We apply methods of machine-learning, such as neural networks, manifold learning and image processing, in order to study 2-dimensional amoebae in algebraic geometry and string theory. With the help of embedding manifold projection, we recover complicated conditions obtained from so-called lopsidedness. For certain cases it could even reach $\sim99\%$ accuracy, in particular for the lopsided amoeba of $F_0$ with positive coefficients which we place primary focus. Using weights and biases, we also find good approximations to determine the genus for an amoeba at lower computational cost. In general, the models could easily predict the genus with over $90\%$ accuracies. With similar techniques, we also investigate the membership problem, and image processing of the amoebae directly.

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arXiv2022

Machine Learning Algebraic Geometry for Physics

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

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arXiv2022

Machine Learning for Hilbert Series

Hilbert series are a standard tool in algebraic geometry, and more recently are finding many uses in theoretical physics. This summary reviews work applying machine learning to databases of them; and was prepared for the proceedings of the Nankai Symposium on Mathematical Dialogues, 2021.

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arXiv2022

Hilbert Series, Machine Learning, and Applications to Physics

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95\%$. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of 'fake' HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered.

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arXiv2022

Machine Learning Calabi-Yau Hypersurfaces

We revisit the classic database of weighted-P4s which admit Calabi-Yau 3-fold hypersurfaces equipped with a diverse set of tools from the machine-learning toolbox. Unsupervised techniques identify an unanticipated almost linear dependence of the topological data on the weights. This then allows us to identify a previously unnoticed clustering in the Calabi-Yau data. Supervised techniques are successful in predicting the topological parameters of the hypersurface from its weights with an accuracy of R^2 > 95%. Supervised learning also allows us to identify weighted-P4s which admit Calabi-Yau hypersurfaces to 100% accuracy by making use of partitioning supported by the clustering behaviour.

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arXiv2021

Polytopes and Machine Learning

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d polytopes with Plücker coordinates as input, which out-perform the usual vertex representation.

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arXiv2021

Some Open Questions in Quiver Gauge Theory

Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories, representation theory, and algebraic geometry. The questions originate from the study of supersymmetric gauge theories in different dimensions with different supersymmetries. Although these constitute merely the tip of a vast iceberg, we hope this guide can give a hint of possible directions in future research. This is an invited contribution to a special volume of Proyecciones, E. Gasparim, Ed., and it is the hope that the questions are specific enough for research projects aimed at PhD students.

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arXiv2021

Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks

We show how to map Grothendieck's dessins d'enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d $\mathcal{N}=2$ supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

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arXiv2021

Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves

We apply machine-learning to the study of dessins d'enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg-Witten curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. The Seiberg-Witten curves for those with rational coefficients are also tabulated.

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arXiv2020

Quiver Mutations, Seiberg Duality and Machine Learning

We initiate the study of applications of machine learning to Seiberg duality, focusing on the case of quiver gauge theories, a problem also of interest in mathematics in the context of cluster algebras. Within the general theme of Seiberg duality, we define and explore a variety of interesting questions, broadly divided into the binary determination of whether a pair of theories picked from a series of duality classes are dual to each other, as well as the multi-class determination of the duality class to which a given theory belongs. We study how the performance of machine learning depends on several variables, including number of classes and mutation type (finite or infinite). In addition, we evaluate the relative advantages of Naive Bayes classifiers versus Convolutional Neural Networks. Finally, we also investigate how the results are affected by the inclusion of additional data, such as ranks of gauge/flavor groups and certain variables motivated by the existence of underlying Diophantine equations. In all questions considered, high accuracy and confidence can be achieved.

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arXiv2020

Lectures on the Calabi-Yau Landscape

In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest in Type II superstring theories, they arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Finally, given the huge amount of Calabi-Yau data, whether and how machine learning can be applied to algebraic geometry and string landscape is also discussed. These notes are directed to the beginning graduate student interested in mathematics and in physics, and are based on lectures given by the 2nd author at the 2019 PIMS Summer School on Algebraic Geometry in High-Energy Physics at the University of Saskatchewan.

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