Our research delves into the rich and intricate landscape of algebraic geometry, centering on the study of vector bundles, principal bundles, and sheaves over a wide variety of algebraic spaces such as projective varieties and algebraic curves. A significant emphasis is placed on understanding their moduli, with particular interest in the notions of stability and semistability. These investigations naturally lead to deeper questions concerning ramified coverings, degeneration phenomena, and the role of singularities in moduli theory. We also explore the structure and action of automorphism groups on moduli spaces and their implications for geometric classification problems.
Beyond these classical themes, our work incorporates techniques from geometric invariant theory, Frobenius splitting, and the study of compactified Jacobians. A distinctive feature of our research is the interplay between characteristic zero and positive characteristic settings, where phenomena often exhibit striking contrasts. We are especially interested in how geometric structures behave under reduction to positive characteristic, and how arithmetic properties influence geometric moduli. Through this blend of foundational theory and advanced techniques, we aim to contribute to a deeper and more unified understanding of modern algebraic geometry.