C0-rigidity of Legendrian submanifolds via microlocal sheaf theory

In this talk, we show that if the image of an embedded Legendrian in a cosphere bundle under a contact homeomorphism is smooth, then it remains Legendrian and its Maslov class is preserved. Our proof is based on the microlocal sheaf theory, using the continuity of the interleaving distance of sheaves with respect to the C0-distance. This is joint work in preparation with Tomohiro Asano, Christopher Kuo, and Wenyuan Li.