Explore a lecture on the construction of a semi-orthogonal decomposition in the derived category of the moduli space of semistable rank-two vector bundles of odd degree on smooth projective curves. Delve into the BGMN Conjecture, examining the use of moduli spaces of stable pairs and GIT wall crossing techniques. Learn about the relationship between derived categories on either side of a given wall through the method of windows. Understand the blocks of the form D(Cd) where Cd are d-th symmetric powers of C, and the conjecturally trivial semi-orthogonal complement to these blocks. Gain insights into this joint work by Sebastián Torres and J. Tevelev, covering topics such as derived categories, direct categories, semistable weights, homomorphisms, and numerical conditions.