arXiv subjects

Xiaoqi Huang

Publications and source records attributed to Xiaoqi Huang.

At least 19 recordsLinked to original sources

arXiv2026

Inhomogeneous Strichartz estimates on manifolds with nonpositive curvature and applications

We prove lossless inhomogeneous Strichartz estimates for solutions to the Schrödinger equation on compact manifolds with nonpositive curvature over frequency-dependent time intervals of length $\log λ\cdot λ^{-1} $. As applications, we improve upon the Sobolev norm growth bounds for the cubic NLS established by Planchon, Tzvetkov and Visciglia on 3-dimensional compact manifolds when the manifold also has nonpositive curvature and extend the lossless homogeneous Strichartz estimates on logarithmic time intervals established by the first author and Sogge to Schrödinger operators with critically singular potentials.

math.AP↗

arXiv2026

A Unified Definition of Hallucination: It's The World Model, Stupid!

Despite numerous attempts at mitigation since the inception of language models, hallucinations remain a persistent problem even in today's frontier LLMs. Why is this? We review existing definitions of hallucination and fold them into a single, unified definition wherein prior definitions are subsumed. We argue that hallucination can be unified by defining it as simply inaccurate (internal) world modeling, in a form where it is observable to the user. For example, stating a fact which contradicts a knowledge base OR producing a summary which contradicts the source. By varying the reference world model and conflict policy, our framework unifies prior definitions. We argue that this unified view is useful because it forces evaluations to clarify their assumed reference "world", distinguishes true hallucinations from planning or reward errors, and provides a common language for comparison across benchmarks and discussion of mitigation strategies. Building on this definition, we also connect our framework to HalluWorld, a complementary benchmark that instantiates fully specified reference world models for stress-testing model hallucinations.

cs.CL↗

arXiv2026

$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $ψ$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $λ$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|ψ\|_{L^6(X)}\lesssim_\varepsilonλ^{\frac{5}{36}+\varepsilon}$ over the local bound $\|ψ\|_{L^6(X)}\lesssimλ^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $ψ$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $ψ$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.

math.NT↗

arXiv2026

Lossless Strichartz and spectral projection estimates on unbounded manifolds

We prove new lossless Strichartz and spectral projection estimates on asymptotically hyperbolic surfaces, and, in particular, on all convex cocompact hyperbolic surfaces. In order to do this, we also obtain log-scale lossless Strichartz and spectral projection estimates on manifolds of uniformly bounded geometry with nonpositive and negative sectional curvatures, extending the recent works of the first two authors for compact manifolds. We are able to use these along with known $L^2$-local smoothing and new $L^2 \to L^q$ half-localized resolvent estimates to obtain our lossless bounds.

math.AP↗

arXiv2025

Weighted geodesic restrictions of arithmetic eigenfunctions

Let $X$ be an arithmetic hyperbolic surface, $ψ$ a Hecke-Maass form, $\ell$ a geodesic segment on $X$, and $μ$ a Borel measure supported on $\ell$ with dimension greater than 1/2. We obtain a power saving over the local bound of Eswarathasan and Pramanik for the $L^2$ norm of $ψ$ with respect to $μ$, which is a weighted generalization of Marshall's geodesic restriction bound and is proved by applying the method of arithmetic amplification. On a general 2-dimensional Riemannian manifold, we also obtain a Kakeya-Nikodym bound for the $L^2$ norm of any Laplace-Beltrami eigenfunction with respect to a Borel measure supported on a geodesic segment with dimension greater than 1/2.

math.NT↗

arXiv2025

Strichartz estimates for the Schrödinger equation on Zoll manifolds

We obtain optimal space-time estimates in $L^q_{t,x}$ spaces for all $q\ge 2$ for solutions to the Schrödinger equation on Zoll manifolds, including, in particular, the standard round sphere $S^d$. The proof relies on the arithmetic properties of the spectrum of the Laplacian on Zoll manifolds, as well as bilinear oscillatory integral estimates, which allow us to relate the problem to Strichartz estimate on one-dimensional tori.

math.AP↗

arXiv2025

Restriction of Schrödinger eigenfunctions to submanifolds

For Schrödinger operators $H_V=-Δ_g+V$ with critically singular potentials $V$ on compact manifolds, we prove sharp estimates for the restriction of eigenfunctions to submanifolds. Our method refines the perturbative argument by Blair-Sire-Sogge and enables us to deal with submanifolds of all codimensions. As applications, we obtain improved estimates on negatively curved manifolds and flat tori. In particular, we extend the uniform $L^2$ restriction estimates on flat tori by Bourgain-Rudnick to singular potentials.

math.AP↗

arXiv2025

Robotic Visual Instruction

Recently, natural language has been the primary medium for human-robot interaction. However, its inherent lack of spatial precision introduces challenges for robotic task definition such as ambiguity and verbosity. Moreover, in some public settings where quiet is required, such as libraries or hospitals, verbal communication with robots is inappropriate. To address these limitations, we introduce the Robotic Visual Instruction (RoVI), a novel paradigm to guide robotic tasks through an object-centric, hand-drawn symbolic representation. RoVI effectively encodes spatial-temporal information into human-interpretable visual instructions through 2D sketches, utilizing arrows, circles, colors, and numbers to direct 3D robotic manipulation. To enable robots to understand RoVI better and generate precise actions based on RoVI, we present Visual Instruction Embodied Workflow (VIEW), a pipeline formulated for RoVI-conditioned policies. This approach leverages Vision-Language Models (VLMs) to interpret RoVI inputs, decode spatial and temporal constraints from 2D pixel space via keypoint extraction, and then transform them into executable 3D action sequences. We additionally curate a specialized dataset of 15K instances to fine-tune small VLMs for edge deployment,enabling them to effectively learn RoVI capabilities. Our approach is rigorously validated across 11 novel tasks in both real and simulated environments, demonstrating significant generalization capability. Notably, VIEW achieves an 87.5% success rate in real-world scenarios involving unseen tasks that feature multi-step actions, with disturbances, and trajectory-following requirements. Project website: https://robotic-visual-instruction.github.io/

cs.RO↗

arXiv2025

Curvature and sharp growth rates of log-quasimodes on compact manifolds

We obtain new optimal estimates for the $L^2(M)\to L^q(M)$, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows $[λ,λ+δ(λ)]$, with $δ(λ)=O((\logλ)^{-1})$ on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$ all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of $L^q$-norms of quasimodes for each Lebesgue exponent $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any $q>q_c$.

math.AP↗

arXiv2024

Weyl laws for Schrödinger operators on compact manifolds with boundary

We prove Weyl laws for Schrödinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has measure 0. These extend the classical results by Seeley, Ivrii and Melrose. The proof uses the Gaussian heat kernel bounds for short times and a perturbation argument involving the wave equation.

math.AP↗

arXiv2024

Strichartz estimates for the Schrödinger equation on compact manifolds with nonpositive sectional curvature

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on compact manifolds with nonpositive sectional curvatures which are related to the classical universal results of Burq, Gérard and Tzvetkov [11]. More explicitly, we are able refine the arguments in the recent work of Blair and the authors [3] to obtain no-loss $L^p_tL^{q}_{x}$-estimates on intervals of length $\log λ\cdot λ^{-1} $ for all {\em admissible} pairs $(p,q)$ when the initial data have frequencies comparable to $λ$, which, given the role of the Ehrenfest time, is the natural analog in this setting of the universal results in [11]. We achieve this log-gain over the universal estimates by applying the Keel-Tao theorem along with improved global kernel estimates for microlocalized operators which exploit the geometric assumptions.

math.AP↗

arXiv2024

Improved spectral projection estimates

We obtain new improved spectral projection estimates on manifolds of non-positive curvature, including sharp ones for relatively large spectral windows for general tori. Our results are stronger than those in an earlier work of the first and third authors [6], and the arguments have been greatly simplified. We more directly make use of pointwise estimates that are implicit in the work of Berard [2] and avoid the use of weak-type spaces that were used in the previous works [6] and [22]. We also simplify and strengthen the bilinear arguments by exploiting the use of microlocal $L^2\to L^{q_c}$ Kakeya-Nikodym estimates and avoiding the of $L^2\to L^2$ ones as in earlier results. This allows us to prove new results for manifolds of negative curvature and some new sharp estimates for tori. We also have new and improved techniques in two dimensions for general manifolds of non-positive curvature.

math.AP↗

arXiv2024

Quasimode concentration on compact space forms

We show that the upper bounds for the $L^2$-norms of $L^1$-normalized quasimodes that we obtained in [9] are always sharp on any compact space form. This allows us to characterize compact manifolds of constant sectional curvature using the decay rates of lower bounds of $L^1$-norms of $L^2$-normalized log-quasimodes fully resolving a problem initiated by the second author and Zelditch [15]. We are also able to characterize such manifolds by the concentration of quasimodes near periodic geodesics as measured by $L^2$-norms over thin geodesic tubes.

math.AP↗

arXiv2024

On Strichartz estimates for many-body Schrödinger equation in the periodic setting

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori $\mathbb{T}^d$, where $d\geq 3$. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in Bourgain-Demeter \cite{BD}. As a comparison, this result can be regarded as a periodic analogue of Hong \cite{hong2017strichartz} though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.

math.AP↗

arXiv2023

Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log λ\cdot λ^{-1} $ for initial data whose frequencies are comparable to $λ$, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We are also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

math.AP↗

arXiv2023

Pointwise Weyl Laws for Schrödinger operators with singular potentials

We consider the Schrödinger operators $H_V=-Δ_g+V$ with singular potentials $V$ on general $n$-dimensional Riemannian manifolds and study whether various forms of pointwise Weyl law remain valid under this pertubation. We prove that the pointwise Weyl law holds for potentials in the Kato class, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and bounded from below or has favorable heat kernel bounds. Moreover, we show that the pointwise Weyl law with the standard sharp error term $O(λ^{n-1})$ holds for potentials in $L^n(M)$.

math.AP↗

arXiv2023

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

The Weyl law of the Laplacian on the flat torus $\mathbb{T}^n$ is concerning the number of eigenvalues $\leλ^2$, which is equivalent to counting the lattice points inside the ball of radius $λ$ in $\mathbb{R}^n$. The leading term in the Weyl law is $c_nλ^n$, while the sharp error term $O(λ^{n-2})$ is only known in dimension $n\ge5$. Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. Moreover, this result verifies the sharpness of the general theorems for the Schrödinger operators $H_V=-Δ_{g}+V$ in the previous work of the authors, and extends the 3-dimensional results of Frank-Sabin to any dimensions.

math.AP↗

arXiv2022

Sharp $L^p$ estimates and size of nodal sets for generalized Steklov eigenfunctions

We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the spectral projection operators on the boundary. Moreover, we derive lower bounds on the size of nodal sets for a variation of the Steklov spectral problem. We consider a generalized version of the Steklov problem by adding a non-smooth potential on the boundary but some of our results are new even without potential.

math.AP↗