D. ArcaraDepartment of Mathematics, Saint Vincent College, Latrobe PA 15650Yuan-Pin LeeDepartment of Mathematics, University of Utah, Salt Lake City, Utah, 84112

We prove that all monomials of κ-classes and ψ-classes are independent in $R^k(\ocM_{g,n})/R^k(\partial\ocM_{g,n})$ for all k≤[g/3]. We also give a simple argument for κ_l≠0 in $R^l(\ocM_g)$ for l≤g−2.

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.

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

We verify the Invariance Conjectures of tautological equations in genus two. In particular, a uniform derivation of all known genus two equations is given.