Our research spans the rich interplay between complex analysis and geometric group theory. A key focus lies in understanding the geometric structures of complex manifolds through intrinsic metrics such as the Kobayashi and Teichmüller metrics. We investigate how Hörmander's $L^2$ estimates for the $\overline{\partial}$-operator offer profound insight into the complex structure of Bers slices and Teichmüller domains. The $\overline{\partial}$-problem acts as a central analytical tool, revealing the subtleties of curvature, completeness, and boundary behavior in complex spaces, and shaping our understanding of deformation theory and quasiconformal mappings in Teichmüller theory.
On a parallel front, our group explores the rich algebraic and topological frameworks underlying braid groups and mapping class groups. We study liftability problems, the Birman-Hilden property, and the algebraic structure of these groups in relation to moduli spaces of curves and surfaces. These investigations uncover links between surface homeomorphisms, unitary representations, and quantum invariants, offering a bridge between low-dimensional topology and complex geometry. By integrating methods from complex analysis and geometric group theory, we aim to illuminate the deep symmetries and fundamental group actions that shape complex surfaces and their moduli.