On the critical LYZ equation in Kähler geometry

In Kähler geometry, the LYZ equation, also known as the deformed Hermitian-Yang-Mills equation, is a fully nonlinear elliptic partial differential equation that originates from mirror symmetry. In this talk, we present the existence of smooth solutions to the LYZ equation at the critical phase, thereby resolving a problem posed by Collins-Jacob-Yau and Y. Li. This is based on joint work with Professors Shing-Tung Yau and Dekai Zhang.