This talk will take place entirely in the algebraic world. We will start with a quick introduction to intersection theory and recall the relevant results from K-theory. The main result is the Grothendieck - Riemann - Roch theorem. Although the theorem is profound the proof is not too difficult so we indicate it. We close by showing that other Riemann-Roch theorems are special cases of this one.

Given a presheaf of spectra F, the problem of descent for F can be divided in two. First, to show that any stably fibrant replacement GF of F is sectionwise stable equivalent to F. Second, to obtain a spectral sequence that compute the sheaf \pi_* (GF) by means of cohomology groups with coefficients in the sheafification of \pi_* F. We will review these notions and some known cases.

In this talk, we will review the proof of the independence of the choice of connections for Chern classes from Chern-Weil theory, which will lead a way to Chern-Simons theory.

This talk will take place entirely in the algebraic world. We will start with a quick introduction to intersection theory and recall the relevant results from K-theory. The main result is the Grothendieck - Riemann - Roch theorem. Although the theorem is profound the proof is not too difficult so we indicate it. We close by showing that other Riemann-Roch theorems are special cases of this one.