One of the cornerstones of geometric analysis is the theory of conformal mappings. In the past century or so, conformal maps have influenced the development of quasiconformal mappings and their generalisations, such as quasisymmetric mappings and, more recently, mappings of finite distortion. The α-Grushin plane, denoted as G_α², is one of the simplest examples of sub-Riemannian manifolds. Recently in [1], a new approach to quasisymmetric maps has been introduced in the setting of the Grushin plane, and later it has been explored in relation to the quasi-conformality in [2]. Moreover, [2] analyses some of the properties of conformal maps in the metric and geometric sense, and [1] introduces a new notion of conformality between domains in the Grushin plane, which has been recently analysed in [3].

During the talk, I will define the Grushin plane and present some of the most interesting results contained in [2] and [3]. In particular, I will introduce a modified system of Beltrami equations, which turns out to be the defining factor for conformal as well as for quasiconformal maps between domains in the Grushin plane.

[1] Ackermann C. An approach to studying quasiconformal mappings on generalized Grushin
planes. Ann Acad Sci Fenn. 2015;
[2] Gartland C, Jung D, Romney M. Quasiconformal mappings on the Grushin plane. Math Z.
2017;
[3] Walicki M. (2024). Conformal mappings on the Grushin plane. Complex Variables and
Elliptic Equations. 2024.