标题： 三叶瓣曲线，散射图，以及辛非压缩性 显示英文标题

标题： Sesquicuspidal curves, scattering diagrams, and symplectic nonsqueezing

摘要： 我们解决了四维椭球到四维圆球的稳定辛嵌入问题。 答案由一个分段光滑函数简洁地编码，该函数表现出从无限斐波那契楼梯到与辛折叠相关的显式有理函数的相变。 我们的方法基于定量辛几何与奇异代数曲线理论之间的桥梁，以及使用散射图来处理这两个主题的一般框架。 特别是，我们构造了复射影平面上具有(p,q)尖点奇点的大量新的有理代数曲线，其中许多曲线解决了具有指定尖点的平面曲线的经典最低次数问题。 Gross--Pandharipande--Siebert的热带顶点群以及对对数卡拉比-丘曲面镜像对称的想法在其中起着关键作用。 我们的许多结果也适用于其他目标空间，例如德尔佩佐曲面和更一般的有理曲面。 显示英文摘要

摘要： We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite Fibonacci staircase to an explicit rational function related to symplectic folding. Our approach is based on a bridge between quantitative symplectic geometry and singular algebraic curve theory, and a general framework for approaching both topics using scattering diagrams. In particular, we construct a large new family of rational algebraic curves in the complex projective plane with a (p,q) cusp singularity, many of which solve the classical minimal degree problem for plane curves with a prescribed cusp. A key role is played by the tropical vertex group of Gross--Pandharipande--Siebert and ideas from mirror symmetry for log Calabi--Yau surfaces. Many of our results also extend to other target spaces, e.g. del Pezzo surfaces and more general rational surfaces.