arXiv subjects
Publications and source records attributed to Zhiyu Liu.
Let $f\colon \big(x\in (X, Δ_X)\big)\to \big(y\in (Y, Δ_Y)\big)$ be a finite surjective morphism between klt singularities such that $K_X+Δ_X=f^*(K_Y+Δ_Y)$. We show that the normalized volumes satisfy \[\widehat{\mathrm{vol}}(x, X, Δ_X)=\mathrm{deg}(f)\cdot \widehat{\mathrm{vol}}(y, Y, Δ_Y).\] This proves a conjecture in [LLX20, Zhu25, XZ26a].
We extend the construction of stability conditions on projective schemes over a field to projective families over an arbitrary base, and prove that they admit proper relative moduli spaces of semistable objects. We also prove a number of complementary results: the existence of mass-Hom bounds for these stability conditions, as conjectured by Halpern-Leistner and Robotis; a comparison with tilt-stability on surfaces and threefolds; a construction of stability conditions on the supported derived category of total spaces of certain vector bundles, including all local Calabi--Yau varieties; and a simple new proof of Bondal and Orlov's reconstruction theorem.
Deep research agents are Large Language Model (LLM)-based systems designed for autonomous, multi-step scientific reasoning, and they hold immense potential for accelerating research in the physical sciences. However, comprehensive and in-depth evaluations of their capabilities within this domain remain lacking. To address this gap, we introduce PhySciBench, a benchmark highly relevant to physical science research, comprising 200 expert-curated questions, balanced between physics and chemistry, across six task categories that reflect real-world scientific workflows. Evaluations of state-of-the-art models and agent systems on PhySciBench reveal limited performance; even the strongest baseline, Gemini Deep Research, achieves an accuracy of only 33.5%. Analysis of failure cases identifies three recurrent deficiencies: fragility in extended reasoning chains, limited knowledge transfer across steps, and a lack of physics-grounded self-verification. Motivated by these findings, we develop DelveAgent, a modular multi-agent framework equipped with an adaptive planning loop, dual-granularity memory, and a hierarchical physics-grounded reflection mechanism. Across four scientific benchmarks, DelveAgent improves accuracy by up to 7.5 percentage points while reducing inference costs to approximately one-third of the strongest baseline. These results establish the significance of PhySciBench as a critical benchmark for evaluating AI systems in the physical sciences and demonstrate that architectural specialization can effectively enhance the reliability of autonomous scientific research. Our data and code are publicly available at https://github.com/yigengjiang/physci-deepresearch.
We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill-Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As an application, we verify Mukai's conjecture on the existence of dual embeddings of $X$. Moreover, we establish a refined categorical Torelli theorem for $X$ and classify autoequivalences of $\mathcal{K}u(X)$. We also give an alternative disproof of Kuznetsov's Fano threefold conjecture.
We generalize the framework of tilt-stability to singular schemes and formulate the generalized Bogomolov-Gieseker inequality conjecture of Bayer-Macrì-Toda for singular threefolds. We also develop relative versions of these constructions, generalizing corresponding results in [BLM+21]. Along the way, we establish Bogomolov-Gieseker-type inequalities for semistable sheaves on any projective scheme. By extending previous techniques, we verify the conjecture for all Fano threefolds with canonical Gorenstein $\mathbb{Q}$-factorial singularities and a series of singular Calabi-Yau threefolds. Furthermore, we construct stability conditions on the relative Kuznetsov components associated with families of singular Fano threefolds, thereby proving a singular analogue of a conjecture of Kuznetsov-Shinder.
Deploying vision-and-language navigation (VLN) agents requires adaptation across diverse scenes and environments, but fine-tuning on a specific scenario often causes catastrophic forgetting in others, which severely limits flexible long-term deployment. We formalize this challenge as the all-day multi-scenes lifelong VLN (AML-VLN) problem. Existing parameter-efficient adapters (e.g., LoRA and its variants) are limited by their two-dimensional matrix form, which fails to capture the multi-hierarchical navigation knowledge spanning multiple scenes and environments. To address this, we propose Tucker Adaptation (TuKA), which represents the multi-hierarchical navigation knowledge as a high-order tensor and leverages Tucker decomposition to decouple the knowledge into shared subspaces and scenario-specific experts. We further introduce a decoupled knowledge incremental learning strategy to consolidate shared subspaces while constraining specific experts for decoupled lifelong learning. Building on TuKA, we also develop a VLN agent named AlldayWalker, which continually learns across multiple navigation scenarios, achieving all-day multi-scenes navigation. Extensive experiments show that AlldayWalker consistently outperforms state-of-the-art baselines.
Traditional language-conditioned manipulation agent sequential adaptation to new manipulation skills leads to catastrophic forgetting of old skills, limiting dynamic scene practical deployment. In this paper, we propose SkillsCrafter, a novel robotic manipulation framework designed to continually learn multiple skills while reducing catastrophic forgetting of old skills. Specifically, we propose a Manipulation Skills Adaptation to retain the old skills knowledge while inheriting the shared knowledge between new and old skills to facilitate learning of new skills. Meanwhile, we perform the singular value decomposition on the diverse skill instructions to obtain common skill semantic subspace projection matrices, thereby recording the essential semantic space of skills. To achieve forget-less and generalization manipulation, we propose a Skills Specialization Aggregation to compute inter-skills similarity in skill semantic subspaces, achieving aggregation of the previously learned skill knowledge for any new or unknown skill. Extensive experiments demonstrate the effectiveness and superiority of our proposed SkillsCrafter.
The problem of low-tubal-rank tensor estimation is a fundamental task with wide applications across high-dimensional signal processing, machine learning, and image science. Traditional approaches tackle such a problem by performing tensor singular value decomposition, which is computationally expensive and becomes infeasible for large-scale tensors. Recent approaches address this issue by factorizing the tensor into two smaller factor tensors and solving the resulting problem using gradient descent. However, this kind of approach requires an accurate estimate of the tensor rank, and when the rank is overestimated, the convergence of gradient descent and its variants slows down significantly or even diverges. To address this problem, we propose an Alternating Preconditioned Gradient Descent (APGD) algorithm, which accelerates convergence in the over-parameterized setting by adding a preconditioning term to the original gradient and updating these two factors alternately. Based on certain geometric assumptions on the objective function, we establish linear convergence guarantees for more general low-tubal-rank tensor estimation problems. Then we further analyze the specific cases of low-tubal-rank tensor factorization and low-tubal-rank tensor recovery. Our theoretical results show that APGD achieves linear convergence even under over-parameterization, and the convergence rate is independent of the tensor condition number. Extensive simulations on synthetic data are carried out to validate our theoretical assertions.
Fix a polarised Calabi-Yau threefold $(X,H)$. We reduce a version of the Bayer-Macrì-Toda conjecture for $(X,H)$, which ensures the existence of Bridgeland stability conditions on $X$, to verifying a Brill-Noether-type inequality for curves on $X$. We then prove this inequality for a broad class of Calabi-Yau threefolds, including complete intersection Calabi-Yau threefolds in weighted projective spaces.
We introduce CMPhysBench, designed to assess the proficiency of Large Language Models (LLMs) in Condensed Matter Physics, as a novel Benchmark. CMPhysBench is composed of more than 520 graduate-level meticulously curated questions covering both representative subfields and foundational theoretical frameworks of condensed matter physics, such as magnetism, superconductivity, strongly correlated systems, etc. To ensure a deep understanding of the problem-solving process,we focus exclusively on calculation problems, requiring LLMs to independently generate comprehensive solutions. Meanwhile, leveraging tree-based representations of expressions, we introduce the Scalable Expression Edit Distance (SEED) score, which provides fine-grained (non-binary) partial credit and yields a more accurate assessment of similarity between prediction and ground-truth. Our results show that even the best models, Grok-4, reach only 36 average SEED score and 28% accuracy on CMPhysBench, underscoring a significant capability gap, especially for this practical and frontier domain relative to traditional physics. The code anddataset are publicly available at https://github.com/CMPhysBench/CMPhysBench.
In this paper, we provide new examples of 1-obstructed and atomic sheaves on an infinite series of locally complete families of projective hyper-Kähler manifolds. More precisely, (1) we prove that the fixed loci of the natural anti-symplectic involutions on the moduli spaces of stable objects in the Kuznetsov component $\mathcal{K}u(X)$ of a Gushel--Mukai fourfold $X$ are 1-obstructed Lagrangian submanifolds, (2) we construct a family of immersed atomic Lagrangian submanifolds on each moduli space of stable objects in $\mathcal{K}u(X)$, and (3) we construct non-rigid projectively hyperholomorphic twisted bundles on any hyper-Kähler manifold of $\mathrm{K3^{[n]}}$-type for infinitely many $n$. Additionally, we discuss examples of atomic Lagrangian submanifolds satisfying $b_1=20$ in a family of hyper-Kähler manifolds of $\mathrm{K3^{[2]}}$-type, as well as atomic sheaves supported on non-atomic Lagrangians.
We consider the noisy matrix sensing problem in the over-parameterization setting, where the estimated rank $r$ is larger than the true rank $r_\star$ of the target matrix $X_\star$. Specifically, our main objective is to recover a matrix $ X_\star \in \mathbb{R}^{n_1 \times n_2} $ with rank $ r_\star $ from noisy measurements using an over-parameterized factorization $ LR^\top $, where $ L \in \mathbb{R}^{n_1 \times r}, \, R \in \mathbb{R}^{n_2 \times r} $ and $ \min\{n_1, n_2\} \ge r > r_\star $, with $ r_\star $ being unknown. Recently, preconditioning methods have been proposed to accelerate the convergence of matrix sensing problem compared to vanilla gradient descent, incorporating preconditioning terms $ (L^\top L + λI)^{-1} $ and $ (R^\top R + λI)^{-1} $ into the original gradient. However, these methods require careful tuning of the damping parameter $λ$ and are sensitive to step size. To address these limitations, we propose the alternating preconditioned gradient descent (APGD) algorithm, which alternately updates the two factor matrices, eliminating the need for the damping parameter $λ$ and enabling faster convergence with larger step sizes. We theoretically prove that APGD convergences to a near-optimal error at a linear rate. We further show that APGD can be extended to deal with other low-rank matrix estimation tasks, also with a theoretical guarantee of linear convergence. To validate the effectiveness and scalability of the proposed APGD, we conduct simulated and real-world experiments on a wide range of low-rank estimation problems, including noisy matrix sensing, weighted PCA, 1-bit matrix completion, and matrix completion. The extensive results demonstrate that APGD consistently achieves the fastest convergence and the lowest computation time compared to the existing alternatives.
We prove a general result on the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations. As an application, we show that the relative Jacobian fibration of cubic fivefolds containing a fixed cubic fourfold can be compactified by a $\mathbb{Q}$-factorial terminal irreducible symplectic variety with the second Betti number at least 24, and admits a Lagrangian fibration whose base is a weighted projective space. In particular, it belongs to a new deformation type of irreducible symplectic varieties.
Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an $8$-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.
In this paper, we conduct the first systematic investigation of twisted cubics on Gushel-Mukai (GM) fourfolds. We then study the double EPW cube, a 6-dimensional hyperkähler manifold associated with a general GM fourfold $X$, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We also prove that a general double EPW cube admits a covering by Lagrangian subvarieties constructed from the Hilbert schemes of twisted cubics on GM threefolds, which provides a new example for a conjecture of O'Grady.
This paper considers the problem of recovering a tensor with an underlying low-tubal-rank structure from a small number of corrupted linear measurements. Traditional approaches tackling such a problem require the computation of tensor Singular Value Decomposition (t-SVD), that is a computationally intensive process, rendering them impractical for dealing with large-scale tensors. Aim to address this challenge, we propose an efficient and effective low-tubal-rank tensor recovery method based on a factorization procedure akin to the Burer-Monteiro (BM) method. Precisely, our fundamental approach involves decomposing a large tensor into two smaller factor tensors, followed by solving the problem through factorized gradient descent (FGD). This strategy eliminates the need for t-SVD computation, thereby reducing computational costs and storage requirements. We provide rigorous theoretical analysis to ensure the convergence of FGD under both noise-free and noisy situations. Additionally, it is worth noting that our method does not require the precise estimation of the tensor tubal-rank. Even in cases where the tubal-rank is slightly overestimated, our approach continues to demonstrate robust performance. A series of experiments have been carried out to demonstrate that, as compared to other popular ones, our approach exhibits superior performance in multiple scenarios, in terms of the faster computational speed and the smaller convergence error.
The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macrì-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field.
We show that a general ordinary Gushel-Mukai(GM) threefold $X$ is reconstructed from the Kuznetsov component $\mathcal{K}u(X)$ together with an extra data coming from tautological sub-bundle of Grassmannian $\mathrm{Gr}(2,5)$. We also prove that $\mathcal{K}u(X)$ determines birational isomorphism class of $X$, while $\mathcal{K}u(X')$ determines the isomorphism class of a general special GM threefold $X'$. As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use $\mathcal{K}u(X)$ to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds.