arXiv subjects

Tony Yue YU

Publications and source records attributed to Tony Yue YU.

8 recordsLinked to original sources

arXiv2026

Higher Embedding Dimension Creates a Stronger World Model for a Simple Sorting Task

We investigate how embedding dimension affects the emergence of an internal "world model" in a transformer trained with reinforcement learning to perform bubble-sort-style adjacent swaps. Models achieve high accuracy even with very small embedding dimensions, but larger dimensions yield more faithful, consistent, and robust internal representations. In particular, higher embedding dimensions strengthen the formation of structured internal representation and lead to better interpretability. After hundreds of experiments, we observe two consistent mechanisms: (1) the last row of the attention weight matrix monotonically encodes the global ordering of tokens; and (2) the selected transposition aligns with the largest adjacent difference of these encoded values. Our results provide quantitative evidence that transformers build structured internal world models and that model size improves representation quality in addition to end performance. We release our metrics and analyses, which can be used to probe similar algorithmic tasks.

cs.LG↗

arXiv2026

Log Calabi-Yau mirror symmetry and non-archimedean disks

We give a uniform construction that includes the mirror algebra to a smooth log Calabi-Yau variety with maximal boundary, proper over an affine variety, or to a one parameter maximal compact Calabi-Yau degeneration, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are counts of non-archimedean analytic disks. More generally, we study the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties.

math.AG↗

arXiv2026

Birational Invariants from Hodge Structures and Quantum Multiplication

We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a non-archimedean version of a non-commutative Hodge structure. The Hodge atoms arise from the spectral decomposition of the F-bundle under the Euler vector field action, and behave additively under blowups, in accordance with Iritani's blowup theorem. We compute several examples and demonstrate applications to birational geometry. In particular, we prove that a very general cubic fourfold is not rational. We also obtain a new proof of the equality of Hodge numbers of birational Calabi-Yau manifolds in any dimension. Furthermore, we show that the framework naturally extends to representations of other motivic Galois groups. This enables the theory of atoms to produce new obstructions to rationality over non-algebraically closed fields of characteristic zero as well.

math.AG↗

arXiv2025

Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants

We establish a comparison result relating non-archimedean cylinder counts and logarithmic cylinder counts in a smooth affine log Calabi-Yau variety. Using the decomposition theorem and the gluing formula from log Gromov-Witten theory, we can express logarithmic cylinder counts in terms of wall type invariants. As a corollary, we show that in the surface case the non-archimedean scattering diagram from Keel-Yu and the logarithmic scattering diagram from Gross-Siebert coincide, and deduce that the two mirror constructions agree. Along the way, we prove the exponential formula, expressing the non-archimedean wall-crossing function as the exponential of a generating series of punctured log Gromov-Witten invariants. This provides the first explicit formula relating counts of non-archimedean curves with boundary to punctured log invariants.

math.AG↗

arXiv2025

Discovering Hidden Algebraic Structures via Transformers with Rank-Aware Beam GRPO

Recent efforts have extended the capabilities of transformers in logical reasoning and symbolic computations. In this work, we investigate their capacity for non-linear latent pattern discovery in the context of functional decomposition, focusing on the challenging algebraic task of multivariate polynomial decomposition. This problem, with widespread applications in science and engineering, is proved to be NP-hard, and demands both precision and insight. Our contributions are threefold: First, we develop a synthetic data generation pipeline providing fine-grained control over problem complexity. Second, we train transformer models via supervised learning and evaluate them across four key dimensions involving scaling behavior and generalizability. Third, we propose Beam Grouped Relative Policy Optimization (BGRPO), a rank-aware reinforcement learning method suitable for hard algebraic problems. Finetuning with BGRPO improves accuracy while reducing beam width by up to half, resulting in approximately 75% lower inference compute. Additionally, our model demonstrates competitive performance in polynomial simplification, outperforming Mathematica in various cases.

cs.LG↗

arXiv2025

Quantum cohomology of variations of GIT quotients and flips

We prove a decomposition theorem for the quantum cohomology of variations of GIT quotients. More precisely, for any reductive group $G$ and a simple $G$-VGIT wall-crossing $X_- \dashrightarrow X_+$ with a wall $S$, we show that the quantum $D$-module of $X_-$ can be decomposed into a direct sum of that of $X_+$ and copies of that of $S$. As an application, we obtain a decomposition theorem for the quantum cohomology of local models of standard flips in birational geometry.

math.AG↗

arXiv2025

Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties

We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.

math.AG↗

arXiv2022

Non-archimedean Gromov-Witten invariants

Motivated by mirror symmetry and the enumeration of holomorphic disks, we construct the theory of Gromov-Witten invariants in the setting of non-archimedean analytic geometry. We build on our previous works on derived non-archimedean geometry and non-archimedean quantum K-invariants, as well as recent developments of rigid analytic motives and virtual fundamental classes in derived geometry. Our approach gives also a new perspective for the classical algebraic case.

math.AG↗