Isoresidual fibration and resonance arrangements
数学学科创建110周年系列报告
报告题目(Title)：Isoresidual fibration and resonance arrangements
报告人(Speaker)：Guillaume Tahar 助理教授 （北京雁栖湖应用数学研究院）
地点(Place)：后主楼1124
时间(Time)：2025年3月14日(周五)，14:00-15:00
邀请人(Inviter)：葛建全
报告摘要
Meromorphic 1-forms on the Riemann sphere, with prescribed orders of singularities, form strata equipped with period coordinates. Fixing the residues at the poles defines the isoresidual fibration of any given stratum onto the vector space of residue configurations. Remarkably, the singular locus of this fibration corresponds to an arrangement of complex hyperplanes. In the specific case of 1-forms with only one zero, the fibration becomes a ramified cover. We provide a formula to compute the degree of this cover and analyze its monodromy. The results leverage the correspondence between complex analysis and the flat geometry of translation surfaces.
The qualitative geometry of these translation surfaces is classified via decorated trees, reducing the computation of the cover's degree to a combinatorial problem. For strata with two zeroes, isoresidual fibers are complex curves endowed with a canonical translation structure. The singularities of these fibers encode, through their local invariants, the corresponding degenerations of the parametrized objects. The monodromy is described in terms of a Gauss-Manin connection, which possesses rich geometric and combinatorial properties. This work is a collaboration with Dawei Chen, Quentin Gendron, and Miguel Prado.
主讲人简介
Tahar主要研究曲面上的几何与动力系统，研究黎曼面的微分形式模空间。在Geometry&Topology, JDE, Illinois J. Math., Letters Math. Physics等著名杂志上发表论文17篇并有9篇在投和6篇准备文章。
