Abstract: This talk will describe some beautiful work of Patrick Iglesias-Zemmour on diffeology. We will start from the definition of diffeological spaces, with some motivations, give some nice properties of diffeological spaces, and introduce diffeological bundles from the point of view of groupoids. Finally, we will discuss the example of irrational torus from a diffeological viewpoint.

Abstract: Given a valuation $v$ on a division algebra $D$, there is an associated graded division algebra $\textrm{gr}(D)$, which is often much easier to work with than $D$ itself, but which retains surprisingly much of the structure of $D$. We illustrate this with $SK_1(D)$. If $v$ on the center of $D$ is Henselian and $D$ is tame, then $SK_1(D)$ is isomorphic to $SK_1(\textrm{gr}(D)).$