Let X be a smooth proper scheme over a field of characteristic 0, and let E be a vector bundle on X. The classical Hirzebruch-Riemann-Roch says that the Euler characteristic of the cohomology H^*(X,E) equals \int_X ch(E) Td(X).

Thus, HRR is an equality of numbers, i.e., elements of a set. In these talks,

we will explain a proof of HRR that uses the hierarchy

{2-categories} -> {1-categories} -> {Vector spaces} -> {Numbers}.

I.e., the origin of HRR will be 2-categorical. The procedure by which we

go down from 2-categories to numbers is that of *categorical trace*.

However, in order to carry out our program, we will need to venture into

the world of higher categories: the 2-category we will be working with

consists of DG-categories, the latter being higher categorical objects.

And the process of calculation of the categorical trace will involve derived

algebraic geometry: the key geometric player will be the self-intersection

of the diagonal of X, a.k.a. the inertia (derived) scheme of X.

So, this series of talks can be regarded as providing a motivation for studying

higher category theory and derived algebraic geometry: we will use them

in order to prove an equality of numbers. That said, we will try to make these

talks self-contained, and so some necessary background will be supplied.