arXiv subjects

Yang-Hui He

Publications and source records attributed to Yang-Hui He.

At least 19 recordsLinked to original sources

arXiv2026

Lipshitz-Sarkar refines mirrors more than Khovanov

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of {\it chat-GTP 5.6 sol Pro} combing over the {\it Regina Census} of knots.

math.GT↗

arXiv2026

CIPro Package: Complete Intersections in Products of Projective Spaces and Line Bundles

CIPro is a Mathematica package for constructing and analyzing complete intersections in products of projective spaces and line bundles over such varieties. It computes properties of complete intersections, such as Chern classes, Hilbert series, GV invariants and symmetries, as well as properties of line bundles on complete intersections, including their cohomology groups. The package also consolidates a number of data sets available in the literature into a single system, including the lists of complete intersection Calabi-Yau three- and four-folds. This short tutorial introduces the package, provides a brief discussion of some of the mathematical background underlying its computations, and gives a series of examples to illustrate its use. These tools are of utility for many computations in string compactifications, especially for Calabi-Yau geometries appearing in Heterotic and Type II constructions. Many tools apply beyond the Calabi-Yau context, including for example, almost Fano bases in F-theory.

hep-th↗

arXiv2026

Computer vision and converse theorems

Random matrices provide a well-established statistical model for a range of arithmetic phenomena. In this paper, we investigate the extent to which one- and two-dimensional convolutional neural networks (CNNs) can distinguish between arithmetic data arising from elliptic curves with conductor in a fixed interval and random matrix data drawn from the same Sato-Tate distribution. Inspired by converse theorems in the Langlands program, we represent each elliptic curve together with its twists as a vector field and, subsequently, encode that vector field as a digital image. We observe that a two-dimensional CNN trained on this image data is better able to separate conductor families from random matrix data than a one-dimensional CNN trained on vectors of Frobenius traces without twisting data. We also observe that the same two-dimensional architecture can predict the analytic rank of an elliptic curve, and it does so by factoring through the (untwisted) Frobenius traces.

math.NT↗

arXiv2026

Murmurations: a case study in AI-assisted mathematics

We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.

math.NT↗

arXiv2026

Phase Transitions in Neural Networks Pruning

Deep neural networks are strongly over-parameterized, often containing far more weights than required for their task. Although such redundancy can aid optimization, it leads to inefficient deployment and high computational cost, motivating model compression techniques. Among these, network pruning provides a clear and effective route to sparsity. We study pruning from a statistical-physics perspective, interpreting performance degradation under weight removal as a phase transition. Focusing on magnitude-based pruning with fine-tuning, we show that deep networks undergo a sharp transition from a cooperative, functional phase to a disordered phase with collapsed performance. This transition is characterized by scaling laws consistent with second-order critical behavior, with connectivity as the control parameter. Our findings suggest universal pruning-induced criticality across architectures and datasets. Finally, we show that there exists a large class of subnetworks sharing the same nodes' degrees with similar learning ability, thus linking model performance to its topological properties.

cond-mat.dis-nn↗

arXiv2025

Machine learning discovers new champion codes

Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains challenging. By training a transformer to predict the minimum Hamming distance of a class of linear codes and pairing it with a genetic algorithm over the search space, we develop a novel method for discovering champion codes. This model effectively reduces the search space of linear codes needed to achieve champion codes. Our results present the use of this method in the study and construction of error-correcting codes, applicable to codes such as generalised toric, Reed-Muller, Bose-Chaudhuri-Hocquenghem, algebrogeometric, and potentially quantum codes.

cs.IT↗

arXiv2025

Mathematics: the Rise of the Machines

We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines in the mathematical sciences. At the heart is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.

math.HO↗

arXiv2025

Rational K3 Homotopy and the Largest Mathieu Group

We interpret the ranks of the rational homotopy groups of a K3 surface as dimensions of representations for the largest sporadic simple Mathieu group. We then construct a vertex algebra equipped with an action by the largest Mathieu group, and use it to associate Jacobi forms to this interpretation, in a compatible way. Our results suggest a topological role for the sporadic simple Mathieu groups in the theory of K3 surfaces.

math.RT↗

arXiv2025

Vacuum Geometry of the Standard Model

Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gr\"obner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gr\"obner bases have doubly exponential computational complexity (for this case, $7^{2^{1023}}$ operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions $1$, $15$, and $29$, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.

hep-th↗

arXiv2025

The Vacuum Moduli Space of the Minimal Supersymmetric Standard Model

A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety $X \subset \mathbb{C}^{49}$ under a symplectic quotient map $\phi$ to $\mathbb{C}^{973}$. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of $\phi$: $\mathbb{C}^{13} \stackrel{\phi^{\texttt{res}}}{\longrightarrow} \mathbb{C}^{22}$. We analyze the geometry of the full vacuum moduli space for superpotentials $W_{\rm minimal}$ (without neutrinos) and $W_{\rm MSSM}$ (with neutrinos) in $\mathbb{C}^{973}$. In both cases, we prove that $X$ consists of three irreducible components $X_1$, $X_2$, and $X_3$, and determine the images $M_i$ of the $X_i$ under $\phi$. For $W_{\rm minimal}$ we show they have, respectively, dimensions $1$, $15$, and $29$, and prove that each of the $M_i$ is a rational variety, while for $W_{\rm MSSM}$ we show that $M_3$ is the only component. Restricting the $M_i$ to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.

hep-th↗

arXiv2025

BPS spectroscopy with reinforcement learning

We apply reinforcement learning (RL) to establish whether at a given position in the Coulomb branch of the moduli space of a 4d $\mathcal{N} = 2$ quantum field theory (QFT) the BPS spectrum is finite. If it is, we furthermore determine the full BPS spectrum at such point in moduli space. We demonstrate that using a RL model one can efficiently determine the suitable sequence of quiver mutations of the BPS quiver that will generate the full BPS spectrum. We analyse the performance of the RL model on random BPS quivers and show that it converges to a solution various orders of magnitude faster than a systematic brute-force scan. As a result, we show that our algorithm can be used to identify all minimal chambers of a given $\mathcal{N}=2$ QFT, a task previously intractable with computer scanning. As an example, we recover all minimal chambers of the $\text{SU}(2)$ $N_f = 4$ gauge theory, and discover new minimal chambers for theories that can be realized by IIB geometric engineering.

hep-th↗

arXiv2024

Metaheuristic Generation of Brane Tilings

The combinatorics of dimer models on brane tilings describe a large class of four-dimensional $\mathcal{N}=1$ gauge theories that afford quiver descriptions and have toric moduli spaces. We introduce a combinatorial optimization method leveraging simulated annealing to explicitly construct geometrically consistent brane tilings, providing a proof of concept for efficient generation of gauge theories using metaheuristic techniques. The implementation of this idea recovers known examples and allows us to derive a new brane tiling with $26$ quantum fields, illustrating the potential of metaheuristic techniques as a valuable addition to the toolbox for constructing and analyzing gauge theories from brane tilings.

hep-th↗

arXiv2024

Futaki Invariants and Reflexive Polygons

Futaki invariants of the classical moduli space of 4d N=1 supersymmetric gauge theories determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4d N=1 supersymmetric gauge theories coming from D3-branes probing Calabi-Yau 3-fold singularities whose bases are Gorenstein Fano surfaces. In particular, we focus on the toric case where the Fano surfaces are given by the 16 reflexive convex polygons and the moduli spaces are given by the corresponding toric Calabi-Yau 3-folds. We study the distribution of and conjecture new bounds on the Futaki invariants with respect to various topological and geometric quantities. These include the minimum volume of the Sasaki-Einstein base manifolds as well as the Chern and Euler numbers of the toric Fano surfaces. Even though the moduli spaces for the family of theories studied are known to be K-stable, our work sheds new light on how the topological and geometric quantities restrict the Futaki invariants for a plethora of moduli spaces.

hep-th↗

arXiv2024

Distinguishing Calabi-Yau Topology using Machine Learning

While the earliest applications of AI methodologies to pure mathematics and theoretical physics began with the study of Hodge numbers of Calabi-Yau manifolds, the topology type of such manifold also crucially depend on their intersection theory. Continuing the paradigm of machine learning algebraic geometry, we here investigate the triple intersection numbers, focusing on certain divisibility invariants constructed therefrom, using the Inception convolutional neural network. We find $\sim90\%$ accuracies in prediction in a standard fivefold cross-validation, signifying that more sophisticated tasks of identification of manifold topologies can also be performed by machine learning.

math.AG↗

arXiv2024

Generating Triangulations and Fibrations with Reinforcement Learning

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

hep-th↗

arXiv2024

A Triumvirate of AI Driven Theoretical Discovery

Recent years have seen the dramatic rise of the usage of AI algorithms in pure mathematics and fundamental sciences such as theoretical physics. This is perhaps counter-intuitive since mathematical sciences require the rigorous definitions, derivations, and proofs, in contrast to the experimental sciences which rely on the modelling of data with error-bars. In this Perspective, we categorize the approaches to mathematical discovery as "top-down", "bottom-up" and "meta-mathematics", as inspired by historical examples. We review some of the progress over the last few years, comparing and contrasting both the advances and the short-comings in each approach. We argue that while the theorist is in no way in danger of being replaced by AI in the near future, the hybrid of human expertise and AI algorithms will become an integral part of theoretical discovery.

math.HO↗

arXiv2024

Learning 3-Manifold Triangulations

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $<8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed.

math.GT↗

arXiv2024

Prolegomena to the Bestiary

``Calabi-Yau Manifolds: a Bestiary for Physicists'' by Tristan Hubsch in 1992 was a classic that served to introduce algebraic geometry to physicists when the first string theory revolution of 1984 - 94 brought, inter alia, the subject of Calabi-Yau manifolds to the staple of high-energy theorists. We are fortunate that a substantially expanded and updated new edition of the Bestiary will shortly appear. This brief note will serve as an afterword to the much anticipated volume.

math.HO↗