Let $k$ be a field, and $A$ be a finite dimensional $k$-algebra. The Auslander-Reiten conjecture (or generalized Nakayama conjecture) says that every finitely generated left $A$-module $M$ satisfying that $\Ext^i_A(M, M\oplus A)=0$ for all $i>0$ must be projective.
The Auslander-Reiten conjecture is still open now, and plays a central role in the representation theory of algebras. And it is closely connected with the celebrated Nakayama conjecture and the finitistic dimension conjecture.
In this talk, we will discuss the behaviors of this conjecture under certain singular equivalences induced by adjoint pairs (joint work with Wei Hu, Yongyun Qin and Ren Wang). As an application, we prove that this conjecture holds for all skew-gentle algebras.