Abstract: Generating functions of intersection numbers of certain tautological classes on moduli spaces of stable curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. Both of them are tau-functions of the KdV hierarchy. Using matrix models, Mironov-Morozov gave a formula expressing Kontsevich-Witten tau function as an expansion of Schur's Q-polynomial with simple coefficients. This formula was called Mironov-Morozov conjecture by Alexandrov. A similar formula was also conjectured by Alexandrov for Brezin-Gross-Witten tau function. In this talk I will describe two proofs of these formulas using Virasoro constraints and cut-and-join operators. The talk is based on joint works with Chenglang Yang.