# Colloquium abstracts

**Sourav Sen**

TIFR

September 8, 2022

**A Cautionary Tale**: Let $ Asubseteq B$ be integral domains and $G$ be a totally ordered Abelian group. D. Daigle has formulated certain hypotheses on degree function $mathrm{deg} : B o Gcup {-infty}$ so that it is tame in characteristic zero, i.e., $mathrm{deg}(D)$ is defined for all $A$-derivations $D : B o B$. This study is important because each $Din mathrm{Der}_k(B)$ for which $mathrm{deg}(D)$ is defined, we can homogenize the derivation which is an important and useful tool in the study of $mathbb{G}_a$-action on an algebraic variety. In arbitrary characteristic, $mathbb{G}_a$-action on an affine scheme $mathrm{Spec}(B)$ can be interpreted in terms of exponential maps on $B$. In this talk we shall discuss analogous formulations of hypotheses on the degree function so that $mathrm{deg}(phi)$ is defined for each $A$-linear exponential map $phi$ on $B$. This talk is based on a joint work with N. Gupta.