A flat homology theory naturally takes values in comodules over a flat Hopf algebroid. In this talk, we will "reverse" this. Starting with a non-flat homology theory, we will construct a new homology theory with values in an abelian category C. Under some conditions, one can show that C is equivalent to the category of comodules of a flat Hopf algebroid. By composing with the forgetful functor to modules, we obtain a new homology theory which is always flat; this is the flat replacement mentioned in the title. The motivating example (due to Piotr Pstragowski) is that complex cobordism is a flat replacement of singular homology with integer coefficients.

Kasparov's KK-theory is a very interesting theory with various applications, such as extensions of $C^*$-algebras and the Novikov conjecture. We will discuss the basic definitions of KK-theory and see how it allows us to recover various properties of K-theory and K-homology such as Bott periodicity, as well as a nice formulation of the Atiyah-Singer index theorem. If time permits, I will also briefly discuss my own interest in KK-theory via the unbounded picture.