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Zelin Jia

Publications and source records attributed to Zelin Jia.

5 recordsLinked to original sources
arXiv2026

On the symplectic forms of Groechenig's Higgs moduli over an elliptic curve

Gorsky, Nekrasov, and Rubtsov introduced the moduli space of marked Higgs bundles over an elliptic curve $E$ and identified it with the Hilbert scheme of points on the cotangent bundle. Later, Groechenig constructed four analogous isomorphisms on marked rational curves, of affine Dynkin types $\widetilde D_4$, $\widetilde E_6$, $\widetilde E_7$, and $\widetilde E_8$, for the $\Gamma$-Hilbert schemes of $T^*E$ for cyclic groups $\Gamma$ with $|\Gamma|\in\{2,3,4,6\}$. We prove that the isomorphisms in all five cases are holomorphic symplectomorphisms.

math.AG↗
arXiv2026

The Logarithmic Asymptotic Phenomenon for Generalized Markov-Hurwitz Equations

The purpose of this paper is twofold. First, we introduce a family of generalized Markov-Hurwitz equations, extending classical Markov-Hurwitz equations with additional degree n-1 interaction terms, Gyoda and Matsushita's generalized Markov equations from 3 variables to n variables. Second, we prove a logarithmic asymptotic phenomenon for the positive integer solutions of these equations.

math.NT↗
arXiv2025

Tropicalization and cluster asymptotic phenomenon of generalized Markov equations

The generalized Markov equations are deeply connected with the generalized cluster algebras of Markov type. We construct a deformed Fock-Goncharov tropicalization for the generalized Markov equations and prove that their tropicalized tree structure is essentially the same as that of the classical Euclid tree. We then define the generalized Euclid tree and prove that it converges to the classical Euclid tree up to a scalar multiple. Moreover, by means of cluster mutations, we exhibit an asymptotic phenomenon, up to some limit q, between the logarithmic generalized Markov tree and the classical Euclid tree. A rationality conjecture of q is then put forward. We also propose a generalized Markov uniqueness conjecture for the generalized Markov equations, which illustrates an application of the asymptotic phenomenon.

math.NT↗
arXiv2024

Dynamic Color Assignment for Hierarchical Data

Assigning discriminable and harmonic colors to samples according to their class labels and spatial distribution can generate attractive visualizations and facilitate data exploration. However, as the number of classes increases, it is challenging to generate a high-quality color assignment result that accommodates all classes simultaneously. A practical solution is to organize classes into a hierarchy and then dynamically assign colors during exploration. However, existing color assignment methods fall short in generating high-quality color assignment results and dynamically aligning them with hierarchical structures. To address this issue, we develop a dynamic color assignment method for hierarchical data, which is formulated as a multi-objective optimization problem. This method simultaneously considers color discriminability, color harmony, and spatial distribution at each hierarchical level. By using the colors of parent classes to guide the color assignment of their child classes, our method further promotes both consistency and clarity across hierarchical levels. We demonstrate the effectiveness of our method in generating dynamic color assignment results with quantitative experiments and a user study.

cs.HC↗