We will discuss the K-theory of complex vector bundles on topological spaces and of holomorphic vector bundles on complex manifolds. A central question is the relationship between K-theory and cohomology. This is done in topology by constructing characteristic classes, but other constructions appear in the holomorphic or algebraic context. We will discuss the Hirzebruch-Riemann-Roch formula, the Atiyah-Hirzebruch spectral sequence, the role of complex cobordism, and other tools developed later on, like the Bloch-Ogus spectral sequence. This talk is part of a subprogram of the Mathematical Science Literature Lecture series, a Memorial Conference for the founders of index theory: Atiyah, Bott, Hirzebruch, and Singer. Talk chair: Baohua Fu Registration：https://harvard.zoom.us/webinar/register/3516062319845/WN_9Qb6IBIfQ0ujujfLvue2eA

Claire Voisin is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of Algebraic Geometry at the Collège de France.