Equivariant moving frames in Cauchy–Riemann geometry

The theory of equivariant moving frames, introduced by Peter Olver in the late 1990s, represents a modern reformulation of Cartan's classical theory of moving frames. Over the last two decades, this constructive theory has demonstrated its large potential for various applications. In particular, recent studies shed light on how the equivariant moving frames techniques are applicable in Cauchy–Riemann ($\mathrm{CR}$) geometry for holomorphic construction of normal forms of real-analytic $\mathrm{CR}$ surfaces and solving their underlying equivalence problems. As the first part of this talk, I will present a brief overview to equivariant moving frames theory. Subsequently and as the second part, I will elaborate on how it facilitates the construction of normal forms for real-analytic manifolds acted on by (pseudo)-groups. I also attempt to illustrate this method by presenting some examples in the context of $\mathrm{CR}$ geometry