Explore the intricate world of Fuchsian groups and parahoric group schemes in this comprehensive lecture from the International Centre for Theoretical Sciences. Delve into key concepts including principal G-bundles, root systems, Cartan matrices, and affine roots. Examine the properties of highest roots in G-simple groups and study the unique example of G2. Investigate root homomorphisms and parahoric groups before focusing on Fuchsian groups and their parabolic representations. Gain insights into advanced mathematical structures and their applications in geometry and topology through this in-depth presentation, complete with a Q&A session to reinforce understanding.