arXiv subjects

Shurui Liu

Publications and source records attributed to Shurui Liu.

7 recordsLinked to original sources

arXiv2026

Kazhdan-Lusztig polynomials of matroids need not be unimodal

We construct, over every finite field, representable matroids whose Kazhdan-Lusztig polynomials are not unimodal. In particular, the conjectures that all Kazhdan-Lusztig polynomials of matroids are log-concave and that they are real-rooted are both false. Our examples are obtained by deleting points from finite projective geometries. More generally, we prove that, under a half-rank degree condition in each contraction quotient, the Kazhdan-Lusztig polynomial of every contraction enumerates the subspaces whose projective points lie in the corresponding deleted set, while the $Z$-polynomial agrees with that of the full projective geometry.

math.CO↗

arXiv2026

Tangent classes of matroids and wonderful compactifications

For every loopless matroid $M$ and every Feichtner--Yuzvinsky building set $\mathcal{G}$ containing the top flat, we construct an integral tangent class $T_{M,\mathcal{G}}^{\mathbb{Z}}\in K_{\mathbb{Z}}(M,\mathcal{G})$; in the realizable case it specializes to the class of the tangent bundle of the corresponding wonderful compactification, it recovers the Hilbert series of the Chow ring through Hirzebruch--Riemann--Roch, and it satisfies the expected Chern-alpha lower bounds. This reproduces the tangent class and its key properties studied by the first author in arXiv:2606.22650. The main body of this paper was produced autonomously, without human mathematical guidance, by Danus, an AI mathematical reasoning agent. Danus solved the problem before arXiv:2606.22650 was publicly available, demonstrating the potential of AI agents in mathematical research. We reproduce its output faithfully, adding only editorial comments; the experiment is documented in Appendix B.

math.AG↗

arXiv2026

Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory

Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.

cs.AI↗

arXiv2026

Automated Conjecture Resolution with Formal Verification

Recent advances in large language models have significantly improved their ability to perform mathematical reasoning, extending from elementary problem solving to increasingly capable performance on research-level problems. However, reliably solving and verifying such problems remains challenging due to the inherent ambiguity of natural language reasoning. In this paper, we propose an automated framework that integrates natural language reasoning with formal verification to tackle research-level mathematical problems. Our framework consists of two components: an informal reasoning agent, Rethlas, and a formal verification agent, Archon. Rethlas combines reasoning primitives with our theorem search engine, Matlas, to explore solution strategies and construct candidate proofs. Archon, equipped with LeanSearch, translates informal arguments into formalized Lean 4 projects through task decomposition, iterative refinement, and automated proof synthesis, ensuring machine-checkable correctness. Using this framework, we resolve an open problem in commutative algebra and formally verify the resulting proof in Lean 4 with essentially no human involvement. Additional case studies illustrate the capabilities of Rethlas in informal mathematical reasoning and discovery, as well as the ability of Archon to formalize research-level proofs in Lean 4. Our experiments demonstrate that strong theorem retrieval tools enable the discovery and application of cross-domain mathematical techniques, while the formal agent can autonomously fill nontrivial gaps in informal arguments. More broadly, our work illustrates a promising paradigm for mathematical research in which informal and formal reasoning systems, equipped with theorem retrieval tools, operate in tandem to produce verifiable results, reduce human effort, and support human-AI collaborative mathematical research.

cs.LG↗

arXiv2026

HiDiGen: Hierarchical Diffusion for B-Rep Generation with Explicit Topological Constraints

Boundary representation (B-rep) is the standard 3D modeling format in CAD systems, encoding both geometric primitives and topological connectivity. Despite its prevalence, deep generative modeling of valid B-rep structures remains challenging due to the intricate interplay between discrete topology and continuous geometry. In this paper, we propose HiDiGen, a hierarchical generation framework that decouples geometry modeling into two stages, each guided by explicitly modeled topological constraints. Specifically, our approach first establishes face-edge incidence relations to define a coherent topological scaffold, upon which face proxies and initial edge curves are generated. Subsequently, multiple Transformer-based diffusion modules are employed to refine the geometry by generating precise face surfaces and vertex positions, with edge-vertex adjacencies dynamically established and enforced to preserve structural consistency. This progressive geometry hierarchy enables the generation of more novel and diverse shapes, while two-stage topological modeling ensures high validity. Experimental results show that HiDiGen achieves strong performance, generating novel, diverse, and topologically sound CAD models.

cs.CV↗

arXiv2025

Higher Period Integrals and Derivatives of L-functions

We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of relative Langlands duality \`a la Ben-Zvi--Sakellaridis--Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral of a Hecke eigensheaf. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin--Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin--Selberg $L$-function. This provides a conceptual generalization of Yun--Zhang's higher Gross--Zagier formula to higher-dimensional spherical varieties.

math.NT↗

arXiv2025

CM-Diff: A Single Generative Network for Bidirectional Cross-Modality Translation Diffusion Model Between Infrared and Visible Images

Image translation is one of the crucial approaches for mitigating information deficiencies in the infrared and visible modalities, while also facilitating the enhancement of modality-specific datasets. However, existing methods for infrared and visible image translation either achieve unidirectional modality translation or rely on cycle consistency for bidirectional modality translation, which may result in suboptimal performance. In this work, we present the bidirectional cross-modality translation diffusion model (CM-Diff) for simultaneously modeling data distributions in both the infrared and visible modalities. We address this challenge by combining translation direction labels for guidance during training with cross-modality feature control. Specifically, we view the establishment of the mapping relationship between the two modalities as the process of learning data distributions and understanding modality differences, achieved through a novel Bidirectional Diffusion Training (BDT). Additionally, we propose a Statistical Constraint Inference (SCI) to ensure the generated image closely adheres to the data distribution of the target modality. Experimental results demonstrate the superiority of our CM-Diff over state-of-the-art methods, highlighting its potential for generating dual-modality datasets.

cs.CV↗