arXiv subjects
Publications and source records attributed to Junyi Xie.
We give a new proof of Faltings' isogeny theorem. More precisely, we show that Yuan's non-archimedean equidistribution theorem can be used to "pump" homomorphisms and semisimplicity from finite fields to number fields, thereby reducing Faltings' theorem directly to Tate's theorem.
Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordinary rational self-map over the function field, but a twisted one. This suggests that twisted rational maps form a natural framework for studying relative dynamics. In this framework we extend the theory of dynamical degrees, the numerical invariants measuring the asymptotic complexity of a dynamical system: the defining limits exist, are independent of the choice of polarization, and are birational invariants. We also identify the relative dynamical degrees of a semi-conjugacy with the dynamical degrees of the induced twisted rational map on the generic fiber, and prove the corresponding mixed degree formula. Finally, using the spectral interpretation of dynamical degrees and valuative methods at infinity, we prove an algebraicity result for the first dynamical degree of twisted endomorphisms of affine varieties.
We prove the Gonality Conjecture in arithmetic dynamics: for any non-isotrivial one-parameter algebraic family of rational maps on $\mathbb{P}^1_{\mathbb{C}}$, the gonality of distinct dynatomic curves tends to infinity. More generally, outside the flexible Latt\`es family, every small sequence of horizontal curves has gonality tending to infinity, and its genus grows superlinearly with its degree over the parameter curve. We also obtain higher-dimensional analogues under natural bifurcation and multiplier-genericity hypotheses. As applications, we prove uniform boundedness results for iterated preimages over number fields and geometric uniform boundedness results for preperiodic points over function fields. The proof combines arithmetic equidistribution, woven currents, and bifurcation theory; the bifurcation mechanism is what forces the growth of genus and gonality.
We survey recent progress on the geometric Bombieri--Lang conjecture over function fields of characteristic zero. We discuss recent work of Xie--Yuan and Guoquan Gao, which together proves the conjecture for varieties admitting finite morphisms to abelian varieties. The guiding idea, developed in joint work with Xinyi Yuan, is that Vojta's dictionary can be made concrete in this setting: from rational points of large height one constructs entire curves on complex fibers.
Let $f,g\in\overline{\mathbb{Q}}[z]$ be polynomials of degree $d\geq2$ with disconnected Julia sets. We prove that they have the same Lyapunov exponent $\mathcal{L}_f=\mathcal{L}_g$ if and only if either $f$ and $g$ are intertwined, or $f$ and $\overline{g}$ are intertwined. The analogous result for critical heights is also obtained. As an application, we provide a new proof of the theorem stating that the multiplier spectrum morphism on the moduli space of polynomials is generically injective.
Sleep staging is essential for sleep assessment and disorder diagnosis. In recent years, automatic sleep staging systems have achieved accuracy approaching that of human experts, but the black-box nature of their predictions hinders clinical adoption. Existing interpretability methods offer partial insight into model behavior, but their outputs still require expert reinterpretation and do not provide a direct basis for auditing individual predictions. To improve trustworthiness, we propose the task of auditable sleep staging. To solve this task, we present SleepVLM, a vision-language model that casts sleep staging as visual reasoning over rendered polysomnography (PSG) waveform images. For each epoch, SleepVLM outputs a stage together with the applicable American Academy of Sleep Medicine (AASM) rules and an auditable rationale. The model is trained using a two-stage framework: Waveform-Perceptual Pre-training followed by Rule-Grounded Supervised Fine-tuning over a mixture of fine-grained and coarse annotations. Experiments on four datasets show that SleepVLM outperforms state-of-the-art methods on average. An automated AASM-feature audit shows broad coverage of stage-defining evidence in the rationales, and independent experts validate their reasoning quality. To facilitate further research, we construct and release MASS-EX, an expert-annotated dataset for rule-grounded sleep staging with AASM rule annotations and expert-written rationales.
We prove the dynamical Mordell-Lang conjecture for product of endomorphisms of an affine curve and a projective curve over $\overline{\mathbb{Q}}$.
We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of $\ell$-primary torsion points on fibers of an abelian scheme over a smooth curve, a result originally proved by Cadoret and Tamagawa. Furthermore, our approach allows us to resolve a conjecture of Cadoret and Tamagawa without additional assumptions. Our approach is based on the theory of Betti foliations and the arithmetic equidistribution theorem.
Cartographic reasoning is the skill of interpreting geographic relationships by aligning legends, map scales, compass directions, map texts, and geometries across one or more map images. Although essential as a concrete cognitive capability and for critical tasks such as disaster response and urban planning, it remains largely unevaluated. Building on progress in chart and infographic understanding, recent large vision language model studies on map visual question-answering often treat maps as a special case of charts. In contrast, map VQA demands comprehension of layered symbology (e.g., symbols, geometries, and text labels) as well as spatial relations tied to orientation and distance that often span multiple maps and are not captured by chart-style evaluations. To address this gap, we introduce FRIEDA, a benchmark for testing complex open-ended cartographic reasoning in LVLMs. FRIEDA sources real map images from documents and reports in various domains and geographical areas. Following classifications in Geographic Information System (GIS) literature, FRIEDA targets all three categories of spatial relations: topological (border, equal, intersect, within), metric (distance), and directional (orientation). All questions require multi-step inference, and many require cross-map grounding and reasoning. We evaluate eleven state-of-the-art LVLMs under two settings: (1) the direct setting, where we provide the maps relevant to the question, and (2) the contextual setting, where the model may have to identify the maps relevant to the question before reasoning. Even the strongest models, Gemini-2.5-Pro and GPT-5-Think, achieve only 38.20% and 37.20% accuracy, respectively, far below human performance of 84.87%. These results reveal a persistent gap in multi-step cartographic reasoning, positioning FRIEDA as a rigorous benchmark to drive progress on spatial intelligence in LVLMs.
Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic $f$-preperiodic points is Zariski-dense in $\mathbb{A}^N$, then some iterate $f^{\circ l}$ ($l\geq1$) is a quotient of a surjective algebraic group endomorphism $g:\mathbb{G}_m^N\to\mathbb{G}_m^N$, over $\overline{\mathbb{Q}}$. This result generalizes a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials to higher dimensions. In fact, we prove a much more general rigidity result for dominant endomorphisms $f$ on an affine variety $X$ defined over a number field, concerning "almost $f$-invariant" Zariski-dense subsets of cyclotomic integral points. We apply our results to backward orbits of regular endomorphisms on $\mathbb{A}^N$ of algebraic degree $d\geq2$, and to periodic points of automorphisms of H\'enon type on $\mathbb{A}^N$.
In this note, we present recent progress on rigidity problems in one-dimensional complex dynamics, including the proof of Dynamical Andr\'e-Oort conjecture for curves and generic injectivity of multiplier spectrum. The proofs combine ideas from algebraic geometry, Arakelov geometry and complex dynamics.
Traditional anomaly detection in human mobility has primarily focused on trajectory-level analysis, identifying statistical outliers or spatiotemporal inconsistencies across aggregated movement traces. However, detecting individual-level anomalies, i.e., unusual deviations in a person's mobility behavior relative to their own historical patterns, within datasets encompassing large populations remains a significant challenge. In this paper, we present BeSTAD (Behavior-aware Spatio-Temporal Anomaly Detection for Human Mobility Data), an unsupervised framework that captures individualized behavioral signatures across large populations and uncovers fine-grained anomalies by jointly modeling spatial context and temporal dynamics. BeSTAD learns semantically enriched mobility representations that integrate location meaning and temporal patterns, enabling the detection of subtle deviations in individual movement behavior. BeSTAD further employs a behavior-cluster-aware modeling mechanism that builds personalized behavioral profiles from normal activity and identifies anomalies through cross-period behavioral comparison with consistent semantic alignment. Building on prior work in mobility behavior clustering, this approach enables not only the detection of behavioral shifts and deviations from established routines but also the identification of individuals exhibiting such changes within large-scale mobility datasets. By learning individual behaviors directly from unlabeled data, BeSTAD advances anomaly detection toward personalized and interpretable mobility analysis.
Inferring demographic attributes such as age, sex, or income level from human mobility patterns enables critical applications such as targeted public health interventions, equitable urban planning, and personalized transportation services. Existing mobility-based demographic inference studies heavily rely on large-scale trajectory data with demographic labels, leading to limited interpretability and poor generalizability across different datasets and user groups. We propose HiCoTraj (Zero-Shot Demographic Reasoning via Hierarchical Chain-of-Thought Prompting from Trajectory), a framework that leverages LLMs' zero-shot learning and semantic understanding capabilities to perform demographic inference without labeled training data. HiCoTraj transforms trajectories into semantically rich, natural language representations by creating detailed activity chronicles and multi-scale visiting summaries. Then HiCoTraj uses a novel hierarchical chain of thought reasoning to systematically guide LLMs through three cognitive stages: factual feature extraction, behavioral pattern analysis, and demographic inference with structured output. This approach addresses the scarcity challenge of labeled demographic data while providing transparent reasoning chains. Experimental evaluation on real-world trajectory data demonstrates that HiCoTraj achieves competitive performance across multiple demographic attributes in zero-shot scenarios.
For endomorphisms of projective varieties, we prove that the arithmetic degree of a point with Zariski dense orbit must be a cohomological Lyapunov multiplier of the dynamical system. We will apply our result to deduce a corollary towards the dynamical Mordell--Lang conjecture.
Psychological support hotlines serve as critical lifelines for crisis intervention but encounter significant challenges due to rising demand and limited resources. Large language models (LLMs) offer potential support in crisis assessments, yet their effectiveness in emotionally sensitive, real-world clinical settings remains underexplored. We introduce PsyCrisisBench, a comprehensive benchmark of 540 annotated transcripts from the Hangzhou Psychological Assistance Hotline, assessing four key tasks: mood status recognition, suicidal ideation detection, suicide plan identification, and risk assessment. 64 LLMs across 15 model families (including closed-source such as GPT, Claude, Gemini and open-source such as Llama, Qwen, DeepSeek) were evaluated using zero-shot, few-shot, and fine-tuning paradigms. LLMs showed strong results in suicidal ideation detection (F1=0.880), suicide plan identification (F1=0.779), and risk assessment (F1=0.907), with notable gains from few-shot prompting and fine-tuning. Compared to trained human operators, LLMs achieved comparable or superior performance on suicide plan identification and risk assessment, while humans retained advantages on mood status recognition and suicidal ideation detection. Mood status recognition remained challenging (max F1=0.709), likely due to missing vocal cues and semantic ambiguity. Notably, a fine-tuned 1.5B-parameter model (Qwen2.5-1.5B) outperformed larger models on mood and suicidal ideation tasks. LLMs demonstrate performance broadly comparable to trained human operators in text-based crisis assessment, with complementary strengths across task types. PsyCrisisBench provides a robust, real-world evaluation framework to guide future model development and ethical deployment in clinical mental health.
We solve the dynamical Mordell-Lang problem in positive characteristic for automorphisms of projective surfaces.
We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (ii) Let $f\times g:X\times C\rightarrow X\times C$ be an endomorphism, where $f$ and $g$ are surjective endomorphisms of a projective variety $X$ and a projective curve $C$, respectively. If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\times C,f\times g)$ have the same form as the return sets of the system $(X,f)$. Using the second result, we deal with the case of split self-maps of products of curves, for which the degrees of the factors are pairwise distinct. In the cases that the height argument cannot be applied, we find examples which show that the return set can be very complicated -- more complicated than experts once imagined -- even for endomorphisms of tori with zero entropy. One may compare them with the conjectures and results stated in [CGSZ21] and [XY25].
Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds). We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of $K3$ surfaces. Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism $f:X\rightarrow X$, all eigenvalues of the action of $f$ on $H^k(X)$ have the same absolute value.