Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112
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.