Explore a detailed calculation of the symmetric and symmetrized topological complexity of the torus in this 33-minute lecture by Isaac Ortigoza. Delve into the completion of these numerical descriptions for closed surfaces, both orientable and non-orientable. Learn about the obstruction theory-based method employed, which hinges on constructing an explicit resolution of integers over the full braid group on two strings for the torus. Gain insights into advanced concepts in applied algebraic topology and expand your understanding of topological complexity in geometric structures.