arXiv subjects

Kaiying Hou

Publications and source records attributed to Kaiying Hou.

8 recordsLinked to original sources

arXiv2026

Towards Autonomous Mathematics Research

Recent advances in foundational models have yielded reasoning systems capable of achieving a gold-medal standard at the International Mathematical Olympiad. The transition from competition-level problem-solving to professional research, however, requires navigating vast literature and constructing long-horizon proofs. In this work, we introduce Aletheia, a math research agent that iteratively generates, verifies, and revises solutions end-to-end in natural language. Specifically, Aletheia is powered by an advanced version of Gemini Deep Think for challenging reasoning problems, a novel inference-time scaling law that extends beyond Olympiad-level problems, and intensive tool use to navigate the complexities of mathematical research. We demonstrate the capability of Aletheia from Olympiad problems to PhD-level exercises and most notably, through several distinct milestones in AI-assisted mathematics research: (a) a research paper (Feng26) generated by AI without any human intervention in calculating certain structure constants in arithmetic geometry called eigenweights; (b) a research paper (LeeSeo26) demonstrating human-AI collaboration in proving bounds on systems of interacting particles called independent sets; and (c) an extensive semi-autonomous evaluation (Feng et al., 2026a) of 700 open problems on Bloom's Erdos Conjectures database, including autonomous solutions to four open questions. In order to help the public better understand the developments pertaining to AI and mathematics, we suggest quantifying standard levels of autonomy and novelty of AI-assisted results, as well as propose a novel concept of human-AI interaction cards for transparency. We conclude with reflections on human-AI collaboration in mathematics and share all prompts as well as model outputs at https://github.com/google-deepmind/superhuman/tree/main/aletheia.

cs.LG↗

arXiv2026

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

cs.AI↗

arXiv2025

$p$-anisotropy on the moment curve for homology manifolds and cycles

We prove that the Gorensteinification of the face ring of a cycle is totally $p$-anisotropic in characteristic $p$. In other words, given an appropriate Artinian reduction, it contains no nonzero $p$-isotropic elements. Moreover, we prove that the linear system of parameters can be chosen corresponding to a geometric realization with points on the moment curve. In particular, this implies that the parameters do not have to be chosen very generically.

math.CO↗

arXiv2025

Bispectral OT: Dataset Comparison using Symmetry-Aware Optimal Transport

Optimal transport (OT) is a widely used technique in machine learning, graphics, and vision that aligns two distributions or datasets using their relative geometry. In symmetry-rich settings, however, OT alignments based solely on pairwise geometric distances between raw features can ignore the intrinsic coherence structure of the data. We introduce Bispectral Optimal Transport, a symmetry-aware extension of discrete OT that compares elements using their representation using the bispectrum, a group Fourier invariant that preserves all signal structure while removing only the variation due to group actions. Empirically, we demonstrate that the transport plans computed with Bispectral OT achieve greater class preservation accuracy than naive feature OT on benchmark datasets transformed with visual symmetries, improving the quality of meaningful correspondences that capture the underlying semantic label structure in the dataset while removing nuisance variation not affecting class or content.

cs.LG↗

arXiv2024

Universal Length Generalization with Turing Programs

Length generalization refers to the ability to extrapolate from short training sequences to long test sequences and is a challenge for current large language models. While prior work has proposed some architecture or data format changes to achieve length generalization, these proposals typically apply to a limited set of tasks. Building on prior scratchpad and Chain-of-Thought (CoT) techniques, we propose Turing Programs, a novel CoT strategy that decomposes an algorithmic task into steps mimicking the computation of a Turing Machine. This framework is both universal, as it can accommodate any algorithmic task, and simple, requiring only copying text from the context with small modifications. We show that by using Turing Programs, we obtain robust length generalization on a range of algorithmic tasks: addition, multiplication and in-context SGD. We then demonstrate that transformers achieve length generalization on random Turing Programs, suggesting that length generalization is possible for any algorithmic task. Finally, we theoretically prove that transformers can implement Turing Programs, constructing a simple RASP (Weiss et al.) program that simulates an arbitrary Turing machine.

cs.LG↗

arXiv2023

Large rank simple bundles of all homological dimensions

For $n\geq 3$ and $r\geq n$, we show that there are rank-$r$ vector bundles on $\mathbb{P}^n$ with arbitrary homological dimension. We apply the Bernstein-Gel'fand-Gel'fand correspondence to translate the vector bundle question into a problem on modules over the exterior algebra. Then, we use linear algebra to construct the desired modules.

math.AG↗

arXiv2023

Learning quantum symmetries with interactive quantum-classical variational algorithms

A symmetry of a state $\vert ψ\rangle$ is a unitary operator of which $\vert ψ\rangle$ is an eigenvector. When $\vert ψ\rangle$ is an unknown state supplied by a black-box oracle, the state's symmetries provide key physical insight into the quantum system; symmetries also boost many crucial quantum learning techniques. In this paper, we develop a variational hybrid quantum-classical learning scheme to systematically probe for symmetries of $\vert ψ\rangle$ with no a priori assumptions about the state. This procedure can be used to learn various symmetries at the same time. In order to avoid re-learning already known symmetries, we introduce an interactive protocol with a classical deep neural net. The classical net thereby regularizes against repetitive findings and allows our algorithm to terminate empirically with all possible symmetries found. Our scheme can be implemented efficiently on average with non-local SWAP gates; we also give a less efficient algorithm with only local operations, which may be more appropriate for current noisy quantum devices. We simulate our algorithm on representative families of states, including cluster states and ground states of Rydberg and Ising Hamiltonians. We also find that the numerical query complexity scales well with qubit size.

quant-ph↗

arXiv2018

The Computational Complexity of Finding Hamiltonian Cycles in Grid Graphs of Semiregular Tessellations

Finding Hamitonian Cycles in square grid graphs is a well studied and important questions. More recent work has extended these results to triangular and hexagonal grids, as well as further restricted versions. In this paper, we examine a class of more complex grids, as well as looking at the problem with restricted types of paths. We investigate the hardness of Hamiltonian cycle problem in grid graphs of semiregular tessellations. We give NP-hardness reductions for finding Hamiltonian paths in grid graphs based on all eight of the semiregular tessilations. Next, we investigate variations on the problem of finding Hamiltonian Paths in grid graphs when the path is forced to turn at every vertex. We give a polynomial time algorithm for deciding if a square grid graph admits a Hamiltonian cycle which turns at every vertex. We then show deciding if cubic grid graphs, even if the height is restricted to $2$, admit a Hamiltonian cycle is NP-complete.

cs.CC↗