Associate Professor He Siqi’s Paper Published Online in Top International Mathematics Journal

Recently, the paper titled “Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification”, co-authored by Associate Prof.He Siqi at the Academy of Mathematics and Systems Science（AMSS），CAS, and Boyu Zhang and Richard Wentworth from the University of Maryland, College Park, was published online in Inventiones Mathematicae.

The study investigates the relationship between an analytic compactification of the moduli space of flat SL(2,C) connections on a closed oriented 3-manifold M (as defined by Taubes) and the Morgan–Shalen compactification of the SL(2,C) character variety of the fundamental group of the manifold. The research establishes an explicit correspondence among Z/2 harmonic 1-forms, measured foliations, and equivariant harmonic maps into R-trees, a correspondence originally proposed by Taubes. As an application, the study proves that on all reducible or Haken manifolds, Z/2 harmonic 1-forms exist for every Riemannian metric. It also demonstrates the existence of manifolds that support singular Z/2 harmonic 1-forms yet have compact SL(2,C) character varieties, thereby disproving a folklore conjecture. In summary, using Z/2 harmonic 1-forms to induce equivariant harmonic maps from the universal cover to R-trees as an intermediate bridge, the study connects and characterizes the Taubes analytic compactification and the Morgan–Shalen compactification, revealing deep interrelations among low-dimensional topology, geometric analysis, and representation theory, thus advancing this interdisciplinary field.

The research was uploaded as a preprint to the arXiv in September 2024, submitted to Inventiones Mathematicae in the same month, formally accepted on May 2, 2026, and is now published online.

He Siqi received his bachelor's degree from the School of Mathematical Sciences at Peking University and got his Ph.D. from the California Institute of Technology in 2018. His primary research interests are differential geometry and topology, focusing on gauge theory and Riemannian manifolds with special holonomy groups, including the Kapustin–Witten equations, Higgs bundles, and calibrated submanifolds.