Higher Derivatives of Operator Functions in Ideals of von Neumann algebras

I will discuss some of my recent work on the problem of differentiating the ``operator function" $\mathcal{I}_{\mathrm{sa}} \ni b \mapsto f(a+b) - f(a) \in \mathcal{I}$, where $\mathcal{I}$ is a certain kind of ideal in a von Neumann algebra $\mathcal{M}$ (for example, $\mathcal{M} = B(H)$ and $\mathcal{I}$ is the space of trace class operators on $H$), $a$ is a self-adjoint operator affiliated with $\mathcal{M}$, and $f \colon \mathbb{R} \to \mathbb{C}$ is an appropriately regular scalar function. Some of the relevant objects are rather technical, so the talk will focus on motivation and exposition rather than precise statements of results.