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We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a
family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.
This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two
degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for
semistable holomorphic bundles. This leads to a natural proof that the hyperk¨ahler Kirwan map is surjective for the non-fixed
determinant case.
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