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Tianyu Yuan

Publications and source records attributed to Tianyu Yuan.

At least 19 recordsLinked to original sources
arXiv2026

Failure-Guided Co-Evolution of Prompts and Training Data

Automatic prompt optimization (APO) improves language-model programs by revising prompts from task feedback, yet it typically holds its training data fixed. Repeatedly optimizing against the same instances confines feedback to weaknesses already represented in those data, leaving related failure conditions unexplored. We therefore view each failure as a dual signal: it indicates both how the prompt should be revised and what new training evidence should be synthesized. We introduce FORGE, a failure-guided framework that co-evolves prompts and training data. FORGE abstracts imperfect executions into reusable failure modes and synthesizes new training data through four complementary mutation strategies. Verified instances are fed back into prompt search, allowing updated prompts to expose the next data needs. Across eight heterogeneous benchmarks, FORGE improves the aggregate score over the unoptimized baseline by 16.52 percentage points and outperforms all evaluated APO baselines. The synthesized data also transfer beyond FORGE: in a transfer study, they improve all nine APO comparisons by 2--9 points and all three GRPO comparisons by 4--8 points under matched optimization budgets. These results establish failures as a shared interface between prompt optimization and data synthesis, and show the benefit of jointly adapting what a model is instructed to do and what it learns from.

cs.SE↗
arXiv2026

Trapezoidality of flat arrangement polynomials

Fox's conjecture predicts that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. Kálmán, Mészáros, and Postnikov gave a new proof of trapezoidality for special alternating links using a polynomial associated with flat vector arrangements. We prove that the flat arrangement polynomial has trapezoidal coefficients for every full-dimensional flat vector arrangement, answering the corresponding open question. Our proof gives a geometric interpretation of this polynomial in terms of convex polytopes. As an application, we prove that the Murasugi--Stoimenow polynomial of every connected Eulerian digraph has trapezoidal coefficients.

math.GT↗
arXiv2026

Post-Training Shifts Confidence: A Three-Stage Analysis of How SFT, RL, and OPD Shape CoT Calibration

Large language models have made strong reasoning gains through supervised fine-tuning, reinforcement learning, and on-policy distillation, yet these post-training methods are usually evaluated only by final-answer accuracy. We study how they reshape confidence during reasoning. We introduce a three-stage calibration framework that evaluates confidence before, during, and after chain-of-thought generation, corresponding to difficulty estimation, early termination, and answer aggregation. Through a controlled comparison on mathematical reasoning benchmarks, we find that OPD provides the most useful pre-reasoning confidence, SFT gives the strongest online signal for early stopping, and RL produces the most reliable trace-level signal for aggregation. We further show that confidence reliability is position-dependent: RL confidence becomes informative after a path-commitment phase, while OPD confidence is useful early but can become inversely calibrated later. Based on this observation, we propose PosConf, a position-aware confidence strategy that uses confidence only from reliable relative-position intervals. PosConf improves RL answer aggregation by 6.1 points over majority voting and consistently improves OPD early stopping under tight token budgets, with gains up to 4.3 points by avoiding its later inverse-calibration region, showing that \emph{confidence in reasoning models should be used both stage-wise and position-awarely}. Our code is available at https://github.com/EIT-NLP/Post-Training-Calibration.

cs.CL↗
arXiv2026

Obstructions to smoothing orbicurves and Orbifold Hecke algebras

In this paper, we study obstructions to smoothing nodal orbicurves with orbighosts mapped into cyclic quotient singularities. As an application, we show, under some assumptions, that the orbifold Hecke algebra of a complex global quotient orbifold $[X/G]$ is isomorphic to the degree $0$ cohomology of the $A_\infty$-algebra of endomorphisms of a regular cotangent fiber $T_{[x]}^*[X/G]$ regarded as an object of the bulk-deformed wrapped Fukaya category of the orbifold $T^*[X/G]$ for a compact complex manifold $X$ and finite group $G \subset \operatorname{Aut}(X)$.

math.SG↗
arXiv2026

Denoise to Track: Harnessing Video Diffusion Priors for Robust Correspondence

In this work, we introduce HeFT (Head-Frequency Tracker), a zero-shot point tracking framework that leverages the visual priors of pretrained video diffusion models. To better understand how they encode spatiotemporal information, we analyze the internal representations of Video Diffusion Transformer (VDiT). Our analysis reveals that attention heads act as minimal functional units with distinct specializations for matching, semantic understanding, and positional encoding. Additionally, we find that the low-frequency components in VDiT features are crucial for establishing correspondences, whereas the high-frequency components tend to introduce noise. Building on these insights, we propose a head- and frequency-aware feature selection strategy that jointly selects the most informative attention head and low-frequency components to enhance tracking performance. Specifically, our method extracts discriminative features through single-step denoising, applies feature selection, and employs soft-argmax localization with forward-backward consistency checks for correspondence estimation. Extensive experiments on TAP-Vid benchmarks demonstrate that HeFT achieves state-of-the-art zero-shot tracking performance, approaching the accuracy of supervised methods while eliminating the need for annotated training data. Our work further underscores the promise of video diffusion models as powerful foundation models for a wide range of downstream tasks, paving the way toward unified visual foundation models.

cs.CV↗
arXiv2026

Higher-dimensional Heegaard Floer homology and spectral networks

Given a closed surface $C$ and a real exact Lagrangian $Σ\subset T^*C$ associated to a spectral curve, we construct a homomorphism $\operatorname{BSk}_κ(C)\to\operatorname{Mat}(N^κ,\operatorname{BSk}_κ(Σ))$ from the braid skein algebra of $C$ to the matrix-valued braid skein algebra of $Σ$ using Floer theory and in particular higher-dimensional Heegaard Floer homology (HDHF). We sketch a proof that this map coincides with a hybrid Floer-Morse approach which counts HDHF-type holomorphic curves coupled with certain Morse gradient graphs -- called fold\-ed Morse trees -- using a variant of the adiabatic limit theorems of Fukaya-Oh and Ekholm, which compares holomorphic curves and Morse flow trees.

math.SG↗
arXiv2025

Heegaard Floer Symplectic homology and Viterbo's isomorphism theorem in the context of multiple particles

Given a Liouville manifold $M$, we introduce an invariant of $M$ that we call the Heegaard Floer symplectic cohomology $SH^*_κ(M)$ for any $κ\ge 1$ that coincides with the symplectic cohomology for $κ=1$. Writing $\hat{M}$ for the completion of $M$, the differential counts pseudoholomorphic curves of arbitrary genus in $\mathbb{R} \times S^1 \times \hat{M}$ that are required to be branched $κ$-sheeted covers when projected to the $\mathbb{R} \times S^1$-direction; this resembles the cylindrical reformulation of Heegaard Floer homology by Lipshitz. These cohomology groups provide a closed-string analogue of higher-dimensional Heegaard Floer homology introduced by Colin, Honda, and Tian. When $\hat{M}=T^*Q$ with $Q$ an orientable manifold, we introduce a Morse-theoretic analogue of Heegaard Floer symplectic cohomology, which we call the free multiloop complex of $Q$. When $Q$ has vanishing relative second Stiefel-Whitney class, we prove a generalized version of Viterbo's isomorphism theorem by showing that the cohomology groups $SH^*_κ(T^*Q)$ are isomorphic to the cohomology groups of the free multiloop complex of $Q$.

math.SG↗
arXiv2025

First-passage time to capture for diffusion in a 3D harmonic potential

We determine the survival probability and first-passage time (FPT) to capture for a harmonically trapped particle, diffusing outside an absorbing spherical boundary by directly solving the differential equation for the survival probability. This solution, obtained as an infinite sum over the relevant eigenfunctions, corrects previously published results [D. S. Grebenkov, J. Phys. A 48, 013001 (2014)]. To verify our calculations, we perform simulations of the survival probability, that accurately reproduce the analytic solutions for a range of parameter values. We then obtain the corresponding FPT distribution as the negative time derivative of the survival probability. Finally, we derive an expression for mean first-passage time (MFPT), also as a sum over eigenfunctions. Numerical evaluation of the first twenty-five terms in this sum closely matches the MFPT obtained by a different method in D. S. Grebenkov, J. Phys. A 48, 013001 (2014). We also find that, in the limit of vanishing trap stiffness, the amplitude of the first term in our infinite-sum solution for the survival probability matches the theoretical escape probability for the potential-free diffusion-to-capture process.

math-ph↗
arXiv2025

Morse theory of loop spaces and Hecke algebras

Given a smooth closed $n$-manifold $M$ and a $κ$-tuple of basepoints $\boldsymbol{q}\subset M$, we define a Morse-type $A_\infty$-algebra $CM_{-*}(Ω(M,\boldsymbol{q}))$, called the based multiloop $A_\infty$-algebra, as a graded generalization of the braid skein algebra due to Morton and Samuelson. For example, when $M=T^2$ the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The $A_\infty$-operations couple Morse gradient trees on a based loop space with Chas-Sullivan type string operations. We show that, after a certain "base change", $CM_{-*}(Ω(M,\boldsymbol{q}))$ is $A_\infty$-equivalent to the wrapped higher-dimensional Heegaard Floer $A_\infty$-algebra of $κ$ disjoint cotangent fibers which was studied in the work of Honda, Colin, and Tian. We also compute the based multiloop $A_\infty$-algebra for $M=S^2$, which we can regard as a derived Hecke algebra of the $2$-sphere.

math.SG↗
arXiv2023

A link invariant from higher-dimensional Heegaard Floer homology

We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration $p\colon W\to D\subset\mathbb{C}$ of a $2n$-dimensional Milnor fiber of the $A_{2κ-1}$ singularity. We represent a link by a $κ$-strand braid, which is expressed as an element $h$ of the symplectic mapping class group $\mathrm{Symp}(W,\partial W)$. We then apply the higher-dimensional Heegaard Floer homology machinery to the pair $(\boldsymbol{a},h(\boldsymbol{a}))$, where $\boldsymbol{a}$ is a collection of $κ$ unstable manifolds of $W$ which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant. This work constitutes part of the author's PhD thesis.

math.SG↗
arXiv2023

Folded Morse flow trees

We present an approach to Morse theory on symmetric products of surfaces using the notion of folded ribbon trees. We introduce an $A_\infty$-category with objects defined as $κ$-tuples of Morse functions, where the differential of the tuple has no self-intersection. We show that when the graph of the differential of the $κ$-tuple of Morse functions on $T^*\mathbb{R}^2$ is the wrapped $κ$ disjoint cotangent fibers, its endormorphism is the Hecke algebra associated to the symmetric group $\mathfrak{S}_κ$.

math.SG↗
arXiv2023

Higher-dimensional Heegaard Floer homology and Hecke algebras

Given a closed oriented surface $Σ$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*Σ$ to the Hecke algebra associated to $Σ$ and show that $\mathcal{F}$ is an isomorphism of algebras. We also establish analogous results for punctured surfaces.

math.SG↗
arXiv2023

The effect of loops on the mean square displacement of Rouse-model chromatin

Many researchers have been encouraged to describe the dynamics of chromosomal loci in chromatin using the classical Rouse model of polymer dynamics by the agreement between the measured mean square displacement (MSD) versus time of fluorescently-labelled loci and the Rouse-model predictions. However, the discovery of intermediate-scale chromatin organization, known as topologically associating domains (TADs), together with the proposed explanation of TADs in terms of chromatin loops and loop extrusion, is at odds with the classical Rouse model, which does not contain loops. Accordingly, we introduce an extended Rouse model that incorporates chromatin loop configurations from loop-extrusion-factor-model simulations. Specifically, we extend the classical Rouse model by modifying the polymer's dynamical matrix to incorporate extra springs that represent loop bases. We also theoretically generalize the friction coefficient matrix so that the Rouse beads with non-uniform friction coefficients are compatible with our Rouse model simulation method. This extended Rouse model allowes us to investigate the impact of loops and loop extrusion on the dynamics of chromatin. We show that loops significantly suppress the averaged MSD of a chromosomal locus, consistent with recent experiments that track fluorescently-labelled chromatin loci in fission yeast [M. L. P. Bailey, I. Surovtsev, J. F. Williams, H. Yan, T. Yuan, S. G. Mochrie, and M. C. King, Mol. Biol. Cell (in press)]. We also find that loops slightly reduce the MSD's stretching exponent from the classical Rouse-model value of 0.5 to a loop-density-dependent value in the 0.45-0.40 range. Remarkably, stretching exponent values in this range have also been reported in recent experiments [S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev. Lett. 104, 238102 (2010) and Bailey et al., Mol. Biol. Cell (in press)].

physics.bio-ph↗
arXiv2022

An example of higher-dimensional Heegaard Floer homology

We count pseudoholomorphic curves in the higher-dimensional Heegaard Floer homology of disjoint cotangent fibers of a two dimensional disk. We show that the resulting algebra is isomorphic to the Hecke algebra associated to the symmetric group.

math.SG↗
arXiv2022

Diffusion of microstructured anisotropic particles in an external field

Microstructured particles are widely used in industries and state-of-the-art research and development. Diffusion of particles, particularly, controlled diffusion by a remotely applied field, has inspired novel applications ranging from targeted drug deliveries, novel procedures for quantifying physical properties of nanoparticles and ambient fluids, to fabrication of composites with enhanced properties. In this work, we report a systematic analysis on field-controlled diffusion of microstructured particles. In account of shape anisotropy and structural heterogeneity of a particle, we study coupled Brownian motions of the particle in $\mathbb{R}^3\times$SO(3). Starting from the microscopic stochastic differential equations of motions, we achieve the coarse-grained Fokker-Planck equation that governs the evolution of the probability distribution function with respect to the position and orientation of the particle. Under some mild conditions, we identify the long-time diffusivity for microstructured particles in an external field. The formulation is applicable to microstructured particles of arbitrary shapes and heterogeneities. As examples of applications, we analyze the diffusion of a heterogeneous spheroidal particle and a pair of spheroidal particles bonded by an elastic ligament. For heterogeneous spheroidal particles, we obtain explicit generalized Stokes-Einstein's relations for diffusivity that accounts for the effects of shape anisotropy, heterogeneity, and an external alignment field. For pairs of spheroidal particles, we consider the superimposed relaxation process from an initial non-equilibrium state to the final equilibrium state. The anomalous scaling of Mean Square Displacement (MSD) with respect to the time of such processes may provide important insight for understanding anomalous diffusions observed in migration of macromolecules and cells in complex viscoelastic media.

cond-mat.soft↗
arXiv2022

A constrained proof of the strong version of the Eshelby conjecture for the three-dimensional isotropic medium

Eshelby's seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the three-dimensional isotropic medium, the weak version of the Eshelby conjecture has been substantiated. The previous work of Ammari et al. substantiates the strong version of the Eshelby conjecture for the cases when the three eigenvalues of the eigenstress are distinct or all the same, whereas the case where two of the eigenvalues of the eigenstress are identical and the other one is distinct remains a difficult problem. In this work, we study the latter case. To this end, firstly, we present and prove a necessary condition for a convex inclusion being capable of transforming a single uniform eigenstress into a uniform elastic stress field. Since the necessary condition is not enough to determine the shape of the inclusion, secondly, we introduce a constraint that is concerned with the material parameters, and prove that there exist combinations of the elastic tensors and uniform eigenstresses such that only an ellipsoid can have the Eshelby uniformity property for these combinations simultaneously. Finally, we provide a more specifically constrained proof of the conjecture by proving that for the uniform strain fields constrained to that induced by an ellipsoid from a set of specified uniform eigenstresses, the strong version of the Eshelby conjecture is true for a set of isotropic elastic tensors which are associated with the specified uniform eigenstresses. This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.

math.AP↗
arXiv2021

Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions

The Eshelby formalism for an inclusion in a solid is of significant theoretical and practical implications in mechanics and other fields of heterogeneous media. Eshelby's finding that a uniform eigenstrain prescribed in a solitary ellipsoidal inclusion in an infinite isotropic medium results in a uniform elastic strain field in the inclusion leads to the conjecture that the ellipsoid is the only inclusion that possesses the so-called Eshelby uniformity property. Previously, only the weak version of the conjecture has been proved for the isotropic medium, whereas the general validity of the conjecture for anisotropic media in three dimensions is yet to be explored. In this work, firstly, we present proofs of the weak version of the generalized Eshelby conjecture for anisotropic media that possess cubic, transversely isotropic, orthotropic, and monoclinic symmetries. Secondly, we prove that in these anisotropic media, there exist non-ellipsoidal inclusions that can transform particular polynomial eigenstrains of even degrees into polynomial elastic strain fields of the same even degrees in them. These results constitute counter-examples, in the strong sense, to the generalized high-order Eshelby conjecture (inverse problem of Eshelby's polynomial conservation theorem) for polynomial eigenstrains in both anisotropic media and the isotropic medium (quadratic eigenstrain only). These findings reveal striking richness of the uniformity between the eigenstrains and the correspondingly induced elastic strains in inclusions in anisotropic media beyond the canonical ellipsoidal inclusions. Since the strain fields in embedded and inherently anisotropic quantum dot crystals are effective tuning knobs of the quality of the emitted photons by the quantum dots, the results may have implications in the technology of quantum information, in addition to in mechanics and materials science.

math-ph↗