This habilitation presents results on two  interconnected moduli-theoretic realms: the geometry of strata of differentials on algebraic curves and the study of representation varieties of fundamental groups, particularly in relation to rigidity and uniformization. Motivated by the dynamics of flat surfaces and their rich geometric structure, the work integrates methods from algebraic geometry, Hodge theory, and differential geometry to address a variety of questions across these domains.

The first part investigates the geometry and topology of strata of strata of k-differentials over curves, focusing on their compactifications, intersection theory, and Chern classes. A first contribution is the detailed analysis of the moduli space of multi-scale differentials—a smooth Deligne–Mumford stack that compactifies the strata in a way amenable to intersection-theoretic techniques. Within this setting, the admissibility of the flat area metric is established, allowing for the computation of Chern classes via Chern–Weil theory. The work also derives explicit and recursive formulas for the orbifold Euler characteristics of strata of Abelian differentials and introduces computational tools, most notably the Sage package diffstrata, to facilitate such calculations.

Subsequent chapters focus on linear submanifolds of strata, encompassing moduli spaces of k-differentials and Teichmüller curves. Their Chern classes are computed, and  Deligne-Mostow ball quotient structures of strata are identified via equality in the Bogomolov–Miyaoka–Yau inequality. In addition, the Kodaira dimension of large families of strata is determined, establishing that several components—including certain spin strata—are of general type in high genus. These results contribute new perspectives on the birational geometry of moduli spaces of differentials and their subvarieties.

The second part of the habilitation addresses rigidity and uniformization phenomena in the context of representations of fundamental groups into Lie groups. We establish a sharp inequality on the gaps between Lyapunov exponents for Hitchin representations, characterizing rigidity through equality. Another contribution extends the Milnor–Wood inequality to varieties with klt singularities, proving that maximal representations arise precisely from finite quotients of ball quotients and remain rigid in this singular setting.