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Ting Gong

Publications and source records attributed to Ting Gong.

At least 19 recordsLinked to original sources
arXiv2026

On potential density of rational points on hyperkahler varieties

In this paper, we prove that the rational points on hyperkähler varieties in each of the four known deformation families admitting a rational Lagrangian fibration are potentially dense. Furthermore, we give several applications and construct examples of hyperkähler varieties beyond these families satisfying the potential density property. Our approach is first to prove that rational points on hyperkähler varieties admitting two rational Lagrangian fibrations with distinct associated isotropic rays are potentially dense. We then construct a projective Shafarevich--Tate twist admitting two such fibrations, together with a dominant rational map from the twist to the original variety.

math.AG↗
arXiv2026

Torelli theorem for the moduli spaces of twisted vector bundles on smooth projective curves

We prove Torelli theorems for moduli space of twisted vector bundles on smooth projective curves over an arbitrary field of characteristic $\neq 2$. A twisted Hitchin construction recovers the curve in higher rank and degree, with partial recovery of the twisting class. For rank two with trivial determinant, we recover the curve and the Brauer class from the moduli stack through an ordered Kummer cover and its polarized Jacobian torsor. For odd determinant on hyperelliptic curves, residual gerbes give a relative Desale--Ramanan construction without an actual determinant line bundle over the ground field. An intrinsic polarized Fano torsor recovers the curve and the class, and a corrected polarization obstruction recovers the determinant gerbe. We also give a twisted Newstead construction in genus two, with an explicit descent condition for its converse, and period--index applications. This is the second part in a series of two papers.

math.AG↗
arXiv2026

Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves

We prove a Torelli theorem for full moduli stacks of vector bundles of rank at least three with fixed determinant on a smooth, proper, geometrically connected, tame Deligne--Mumford curve of coarse genus at least two over an arbitrary field $k$. We also prove a Torelli theorem for the full moduli stack of vector bundles of rank at least three of fixed-degree under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. This is the first part in a series of two papers.

math.AG↗
arXiv2026

On Intersections of System Normalizers

We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\geq1$, we construct a finite metabelian group $G_{m,q}$ with $Z_\infty(G_{m,q})=1$ and $|Φ(G_{m,q})|=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number is unbounded even when the order of the Frattini subgroup is fixed.

math.GR↗
arXiv2026

An order automorphism of a Dlab group not induced by conjugation

Let $G=D_{\langle2\rangle}([0,1])$ be equipped with either of its two Dlab orders. There exists an order automorphism $α$ of $G$ such that, for every rank-one subgroup $H\leq\mathbb R_{>0}^{\times}$, every one of the six corresponding Dlab groups $A$ listed below, equipped with any of its linear orders, every order-preserving embedding $e:G\hookrightarrow A$, and every $u\in A$, there exists $f\in G$ such that $e(α(f))\neq u^{-1}e(f)u$.

math.GR↗
arXiv2026

Total 3-closure for projective special linear groups

A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.

math.GR↗
arXiv2026

Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.

math.GR↗
arXiv2026

Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration

Large language models can contribute useful ideas to mathematical research, yet long-horizon proof attempts remain difficult to coordinate, evaluate, and reproduce. We present Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management. We evaluate Albilich on the RealMath benchmark (Zhang et al. 2025) and on open problems in group theory from the Kourovka Notebook (Khukhro and Mazurov 2026). It solved 10/10 problems on RealMath with CAS and 9/10 with no CAS. On the Kourovka problems, Albilich produced a counterexample to Problem 21.142 and a proof of a strengthening of Problem20.2. Anablation on Problem 17.91 demonstrates 32.0% token reduction when CAS is enabled. An ablation on Problem 21.142 demonstrates higher verifier-rejection rate and failure to synthesize proof routes in the absence of the advisor agent. These results support Albilich as a human-steerable, CAS-boosted environment for scalable AI-assisted mathematical research.

cs.AI↗
arXiv2026

TwistedMerge: Certified Higher-Order Diagnostics and Abstention for Model Merging

Model merging combines independently trained or fine-tuned models, but pairwise alignability does not imply globally consistent alignment. We formulate merging as a finite descent problem in which checkpoints are local objects, alignment maps are transitions, and cycle products are residuals. TwistedMerge is a conservative certification pipeline that separates fixed-chart averaging, synchronization-removable gauge inconsistency, a certified central obstruction on a specified comparison complex, and nonabelian holonomy. A residual is promoted to a cohomology class only after inverse-consistency, coefficient-identification, centrality, and closure tests; otherwise the method abstains and returns an ordinary or synchronized fallback. We prove a constant-edge no-go result, frozen-complex three-way and predeclared-family error-control theorems, and a refinement test for comparison-complex sensitivity. A planted neural alignment defect is removed by cycle-consistent synchronization, showing that a nonzero cycle score alone is not a higher obstruction. Controlled central systems recover the predicted non-coboundary and projective-rank behavior, while noisy estimates move from certification to abstention without false lifts on the tested controls. A trained low-rank-adapter audit shows that naive factor averaging depends on the chosen GLr representative, whereas global factor synchronization and dense-delta SVD are stable. On natural checkpoint collections, cycle residuals do not predict merge degradation and no natural central or period-index class is certified. The results position descent theory as a falsifiable certification and abstention framework.

cs.LG↗
arXiv2026

Timestep Rescheduling in Diffusion Inversion

Diffusion inversion, which maps images back to the Gaussian latent space of a diffusion model, is a critical task for image reconstruction and editing. While DDIM enables fast deterministic inversion, it inherently introduces deviations that accumulate into noticeable inversion errors. Existing methods often address this by solving a fixed-point problem but largely overlook how the selection of the diffusion timestep in the noise scheduler influences inversion fidelity. In this work, we reveal that the deviation scale in diffusion inversion is strongly dependent on the timestep size, and exhibits a parabolic trend, with larger errors concentrated at both small and large timesteps. Based on this finding, we propose a simple yet effective nonuniform timestep scheduler that integrates a global rescaling with a local dynamic programming based rescheduling, enabling a strategic allocation of computational effort that minimizes the overall inversion error and preserves higher inversion accuracy. Our method serves as an off-the-shelf enhancement for existing inversion techniques and requires no extra parameters or computational overhead. Through extensive experiments, we verify that integrating our scheduler consistently boosts the performance of existing inversion methods, achieving superior results in image reconstruction and editing.

cs.CV↗
arXiv2026

Slogans or Stance? A Label-Light Diagnostic for Entrepreneurial-Discourse Measurement on Chinese SOE Speeches

Dictionary methods, topic models, and embedding-similarity scorers are widely used in CSS and management research to measure constructs such as "entrepreneurial spirit" in corporate speeches. We contribute a label-light measurement diagnostic for such instruments rather than a new extraction model. On a corpus of 80 speeches by leaders of centrally administered Chinese state-owned enterprises, we exploit a natural experiment of 24 same-company different-speaker pairs and 5 same-company same-speaker pairs to test whether a method's per-document indices vary with leader identity holding firm constant. LDA fails (Cohen d=0.20, 95% CI [-0.72, 1.20]); a dictionary scorer reaches d=0.81 and a Chinese sentence encoder d=0.65 on doc-vector distances of order 10^-3. A zero-shot 9B open-weight LLM (Qwen3.5:9b) raises paired-contrast d to 1.09 (exact permutation p1=0.034). We downgrade three claims accordingly: gold F1 measures consistency with the LLM's own prompt rule rather than external construct recovery; doc-level style residualisation cuts the LLM's d to 0.43 (p1=0.22), so roughly half of the effect is consistent with leader idiolect; and a confidence-weighted calibration trades Delta for variance with an auto-mined slogan lexicon near-inert in ablation. We release the 2,190-segment scored corpus, the 170-paragraph pilot, the slogan lexicon, two-family LLM scores, and the evaluation harness.

cs.CL↗
arXiv2026

Moduli of vector bundles on $μ_n$-gerbes over genus 2 curves and the period-index problem

We develop a framework for describing vector bundles on $μ_n$-gerbes over curves and illustrate the construction through two detailed examples. Using the interpretation of Brauer classes as obstructions to descending determinantal line bundles from the algebraic closure, together with a geometric analysis of the moduli space of twisted sheaves, we prove that for genus $2$ curves there exist Brauer classes over the base field whose period equals their index. Over $C_1$-fields, we further show that every $2$-torsion class in the Brauer group of a genus $2$ curve satisfies the period-index problem. As an application, we construct higher-dimensional varieties obtained as fibre products of genus $2$ curves over $C_1$-fields whose $2$-torsion algebraic Brauer classes also satisfy the period-index problem, providing new evidence toward the period-index conjecture.

math.AG↗
arXiv2025

Picard group action on the category of twisted sheaves

In this paper, we study the category of twisted sheaves over a scheme $X$. Let $\mathcal{M}$ be a quasi-coherent sheaf on $X$, and $α$ in $\operatorname{Br}(X)$. We show that the functor $ - \otimes_{\mathcal{O}_X} \mathcal{M} : \operatorname{QCoh}(X, α) \to \operatorname{QCoh}(X, α) $ is naturally isomorphic to the identity functor if and only if $\mathcal{M}\cong \mathcal{O}_{X}$. As a corollary, the action of $\operatorname{Pic}(X)$ on $D^{b}(X, α)$ is faithful for any Noetherian scheme $X$.

math.AG↗
arXiv2025

PRIME: Phase Reversed Interleaved Multi-Echo acquisition enables highly accelerated distortion-free diffusion MRI

Purpose: To develop and evaluate a new pulse sequence for highly accelerated distortion-free diffusion MRI (dMRI) by inserting additional echoes without prolonging TR, when generalized slice dithered enhanced resolution (gSlider) radiofrequency encoding is used for volumetric acquisition. Methods: A phase-reversed interleaved multi-echo acquisition (PRIME) was developed for rapid, high-resolution, and distortion-free dMRI, which includes several echoes where the first echo is for target diffusion-weighted imaging (DWI) acquisition with high-resolution and additional echoes are acquired with either lower resolution for 1) high-fidelity field map estimation, 2) phase navigation for shot-to-shot phase correction, 3) motion navigation across diffusion directions, or with high resolution to enable 4) high fidelity diffusion relaxometry acquisitions. The sequence was evaluated on in vivo data acquired from healthy volunteers on clinical and Connectome 2.0 scanners. Results: In vivo experiments demonstrated that 1) high in-plane acceleration (Rin-plane of 5-fold with 2D partial Fourier) was achieved using the high-fidelity field maps estimated from the second echo, which was made at a lower resolution/acceleration to increase its SNR while matching the effective echo spacing of the first readout, 2) high-resolution diffusion relaxometry parameters were estimated from triple-echo PRIME data using a white matter model of multi-TE spherical mean technique (MTE-SMT), and 3) high-fidelity mesoscale DWI at 490 um isotropic resolution was obtained in vivo by capitalizing on the high-performance gradients of the Connectome 2.0 scanner. Conclusion: The proposed PRIME sequence enabled highly accelerated, high-resolution, and distortion-free dMRI using additional echoes without prolonging scan time when gSlider encoding is utilized.

physics.med-ph↗
arXiv2025

Interplay between MRI-based axon diameter and myelination estimates in macaque and human brain

Axon diameter and myelin thickness affect the conduction velocity of action potentials in the nervous system. Imaging them non-invasively with MRI-based methods is thus valuable for studying brain microstructure and function. Electron microscopy studies suggest that axon diameter and myelin thickness are closely related to each other. However, the relationship between MRI-based estimates of these microstructural measures, known to be relative indices, have not been investigated across the brain mainly due to methodological limitations. In recent years, studies using ultra-high gradient strength diffusion MRI (dMRI) have demonstrated improved estimation of axon diameter index across white-matter (WM) tracts in the human brain, making such investigations feasible. In this study, we aim to investigate relationships between tissue microstructure properties across white-matter tracts, as estimated with MRI-based methods. We collected dMRI with ultra-high gradient strength and multi-echo spin-echo MRI on ex vivo macaque and human brain samples on a preclinical scanner. From these data, we found that the correlations between axon diameter index and other microstructural imaging parameters were weak but consistent across WM tracts in samples estimated with sufficient signal to noise ratio. In well-myelinated regions, tissue voxels with larger axon diameter indices were associated with lower packing density, lower MWF and a tendency of higher g-ratio. We also found that intra-axonal signal fractions and MWF were not consistently correlated when assessed in different samples. Overall, the findings suggest that MRI-based axon geometry and myelination measures can provide complementary information about fiber morphology, and the relationships between these measures agree with prior electron microscopy studies in smaller field of views.

physics.med-ph↗
arXiv2025

Microstructure.jl: a Julia Package for Probabilistic Microstructure Model Fitting with Diffusion MRI

Microstructure.jl is a Julia package designed for probabilistic estimation of tissue microstructural parameters from diffusion or combined diffusion-relaxometry MRI data. It provides a flexible and extensible framework for defining compartment models and includes robust and unified estimators for parameter fitting and uncertainty quantification. The package incorporates several established models from the literature, such as the spherical mean technique and soma and neurite density imaging (SANDI), along with their extensions for analyzing combined diffusion and T2 mapping data acquired at multiple echo times. For parameter estimation, it features methods like Markov Chain Monte Carlo (MCMC) sampling and Monte Carlo dropout with neural networks, which provide probabilistic estimates by approximating the posterior distributions of model parameters. In this study, we introduce the major modules, functionality, and design of this package. We demonstrate its usage in optimizing acquisition protocols and evaluating model fitting performance with synthesized datasets. We also showcase practical applications with publicly available datasets. Microstructure.jl is applicable to in vivo and ex vivo imaging data acquired with typical research, high-performance, or pre-clinical scanners.

physics.med-ph↗
arXiv2024

Compartment-specific estimation of T2 and T2* with diffusion-PEPTIDE MRI

We present a microstructure imaging technique for estimating compartment-specific T2 and T2* simultaneously in the human brain. Microstructure imaging with diffusion MRI (dMRI) has enabled the modelling of intra-neurite and extra-neurite diffusion signals separately allowing for the estimation of compartment-specific tissue properties. These compartment-specific properties have been widely used in clinical studies. However, conventional dMRI cannot disentangle differences in relaxations between tissue compartments, causing biased estimates of diffusion measures which also change with TE. To solve the problem, combined relaxometry-diffusion imaging methods have been developed in recent years, providing compartmental T2-diffusion or T2*-diffusion imaging respectively, but not T2 and T2* together. As they provide complementary information, a technique that can estimate both jointly with diffusion is appealing to neuroimaging studies. The aim of this work is to develop a method to map compartmental T2-T2*-diffusion simultaneously. Using an advanced MRI acquisition called diffusion-PEPTIDE, a novel microstructure model is proposed and a multi-step fitting method is developed to estimate parameters of interest. We demonstrate for the first time that compartmental T2, T2* can be estimated simultaneously from in vivo data. we further show the accuracy and precision of parameter estimation with simulation.

physics.med-ph↗