Ordinary homology is a geometrically defined invariant of spaces: the 0-th homology group counts the number of components; the n-th homology group counts n-cycles, which correspond to an intuitive notion of 'n-dimensional holes' in a space. K-theory, or more specifically the 0-th K-theory group, is defined in terms of vector bundles, and so also has an immediate relationship to geometry. By contrast, the n-th K-theory group is typically defined homotopy-theoretically using the black box of Bott periodicity.
I will describe a more geometric perspective on K-theory, using Z/2-graded vector bundles and bundles of modules for Clifford algebras. Along the way I will explain Clifford algebras, 2-categories, and Morita equivalence, explicitly check the purely algebraic 8-fold periodicity of the Clifford algebras, and discuss how and why this periodicity implies Bott periodicity.
The talk will not presume any prior knowledge of K-theory, Clifford algebras, Bott periodicity, or the like.
Based on joint work with Arthur Bartels and Andre Henriques