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In "Some numerical results in complex differential geometry," Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (Y. Sano, "Numerical algorithm for finding balanced metrics"). In "Numerical solution to the Hermitian Yang-Mills equation on the Fermat quintic," Douglas, et. al., conjecture that the same holds in the case of vector bundles. In this paper, we give an affirmative answer to their conjecture.
We introduce the notion of tautological ring for the moduli space of spin curves. Moreover, we study some relations among tautological classes which are motivated by physics. Finally, we show that the Chow rings of these moduli spaces are tautological in low genus.