  The main goal of this paper is to prove the following two conjectures for genus up to two:   1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy.   2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds.   The main technique used in the proof is the invariance of tautological equations under loop group action. 
Witten's conjecture, Virasoro conjecture, and invariance of tautological   equations