The Harder-Narasimhan Filtration and Stability Conditions for Vector Bundles

Explore a mathematical seminar talk that delves into the stability condition for vector bundles and the Harder-Narasimhan filtration. Learn how D. Mumford introduced the concept of slope stability, which despite its initially unintuitive nature, proves crucial in Geometric Invariant Theory and moduli problems. Discover how the Harder-Narasimhan filtration provides a canonical method for describing vector bundles on curves through stable bundle extensions. Follow the generalization of this theory to reductive groups by A. Ramananthan and K. Behrend using complementary polyhedra, with specific focus on GL2 over P^1 as an illustrative example of the equivalence between Ramananthan-Behrend and slope stability. Gain insights into S. Schieder's slope map as a third perspective on stability, particularly in its role clarifying the relationship between different stability notions in the GLn case.