Two talks, note the unusual time and place:
We start with the definition of a cone over a Deligne-Mumford stack and, for cones equipped with an action of a vector bundle, we construct the corresponding quotient stack. A cone stacks over a Deligne-Mumford stack is defined to be an Artin stack, which is locally a quotient of a cones by a vector bundle.
A key example of these constructions is the following. Let $i:X \hookrightarrow Y$ be a closed immersion, the normal cone ${\mathcal C}_{X/Y}$ of $i$ is equipped with an action of the tangent bundle $i^* T_Y$.
The intrinsic normal cone of a Deligne-Mumford stack $\mathcal{X}$ is a cone stack defined étale-locally by quotients of the form $[{\mathcal C}_{X/Y}/ i^* T_Y].
The intrinsic normal cone of a Deligne-Mumford stack can be also defined using the cotangent complex, we explain briefly this approach.
Finally, we describe the relationship between the intrinsic normal sheaf of a Deligne-Mumford stack and the deformations of an affine scheme over it, showing that the normal sheaf carries the obstructions for these deformations.