广义簇代数的散射图

Scattering diagrams, Hall algebras and stability conditions

To any quiver with relations we associate a consistent scattering diagram taking values in the motivic Hall algebra of its category of representations. We show that the chamber structure of this scattering diagram coincides with the natural chamber structure in an open subset of the space of stability conditions on the associated triangulated category. In the three-dimensional Calabi-Yau situation, when the relations arise from a potential, we can apply an integration map to give a consistent scattering diagram taking values in a tropical vertex group.

秩大于5的拟仿射广义Dynkin图的分类

Classification of quasi-affine Generalized Dynkin Diagrams with Rank $> 5$

本文分类了所有秩大于5的拟仿射连通广义Dynkin图，并进一步确定了相应的拟仿射Nichols（Lie辫）代数。

散射图：多项式性与稠密区域

Scattering diagrams: polynomiality and the dense region

本文通过散射图的形变与变异，证明其系数在参数上呈多项式性且于稠密区域非平凡，并探讨了对 Gromov-Witten 不变量与箭图表示的推论。

秩为 $5$ 的拟仿射广义 Dynkin 图的分类

Classification of quasi-affine Generalized Dynkin Diagrams with Rank $ 5$

该研究完成秩 $5$ 情形的分类，确定全部拟仿射连通广义 Dynkin 图及相应拟仿射 Nichols（Lie 辫子）代数。

Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.

镜像对称中的散射图

Scattering diagrams in mirror symmetry

本文介绍散射图在镜像对称中的核心作用，阐述其与SYZ猜想和HMS猜想的关联，旨在理解重构问题。

散射图、紧致分次与广义正性

Scattering diagrams, tight gradings, and generalized positivity

本文提出了一类新的组合对象“紧致分次”，借此给出了描述秩2一致散射图的正性公式，并证明了任意秩广义 cluster 散射图的墙函数系数为正，从而确立了广义 cluster algebras 的 Laurent 正性及其 theta 基的强正性。

稳定性散射图与箭图覆盖

Stability scattering diagrams and quiver coverings

本文证明带势箭图覆盖的稳定性散射图在良好分次下通过限制操作关联，并应用于带标记曲面，得出单穿孔闭曲面的手镯基与theta基一致，且稳定性散射图与簇散射图一致（除单穿孔环面外）。

秩2聚类散射图中低次的显式形式

Explicit forms in lower degrees of rank 2 cluster scattering diagrams

本文基于双对数元素研究秩2聚类散射图的墙元素，推导出计算低次墙元素的方法，并给出Badlands的显式形式，同时发现与次数无关的墙结构。

散射图与Jeffrey-Kirwan留数

Scattering diagrams and Jeffrey-Kirwan residues

本文利用Maurer-Cartan渐近分析，证明了初始散射图的一致完全化及其theta函数可由显式亚纯形式的Jeffrey-Kirwan留数表示。

散射扇

Scattering fans

研究散射图与簇代数的关系，证明簇散射扇细化突变扇，并猜想有限突变类型的交换矩阵对应的两个扇形一致。

Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations

We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Carta…

Scattering diagrams, theta functions, and refined tropical curve counts

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture o…

Scattering diagrams, sheaves, and curves

We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on $\mathbb{P}^2$. This survey is based on…

$\u21d2$射影平面$\u21d0$的稳定性散射图几何及其应用

Geometry of the stability scattering diagram for $\mathbb{P}^2$ and applications

本研究详细分析了Bousseau提出的射影平面$\u21d2$的稳定性散射图，将其划分为三个区域并揭示了各区域结构与特性，证明有界区域上边界为Le Potier曲线，并给出了模空间墙交叉的应用。