Abstract. We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.

Abstract. We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2, R) in degree four, establishing a special case of a conjecture of Monod.

We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite center, the same argument shows that the image of the comparison map contains all the even generators. Our proof uses a Hirzebruch type proportionality principle in combination with Gromov's results on boundedness of primary characteristic classes and classical results of Cartan and Borel on the cohomology of compact homogeneous spaces. orem 1 we learned from Michelle Bucher-Karlsson that a similar theorem should be true for certain even-degree classes in arbitrary semisimple Lie groups. We are indepted to her for this suggestion, which led us to discover Theorem 2. The second author would like to thank the Department of Mathematics at Rutgers University for their hospitality and excellent working conditions. Outline of the proofIn this section we outline the proofs of Theorem 1 and Theorem 2.Let G be an arbitrary semisimple Lie group with finite center and without compact factors. The main technical result of this article is Proposition 2.2 below,

We analyze existence, uniqueness and regularity of solutions for perturbations of the Spence-Abel equation for the Rogers' dilogarithm. As an application we deduce a version of Hyers-Ulam stability for the Spence-Abel equation. Our analysis makes use of a well-known cohomological interpretation of the Spence-Abel equation and is based on our recent results on continuous bounded cohomology of SL 2 (R).

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2, R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles. defined by L κ (α) = κ α.Corollary. The bounded Lefschetz map in (1) is zero in all positive degrees.Returning to Theorem 1, we note that in small degrees n = 3, 4, much stronger vanishing theorems apply: Burger and Monod [17] proved that H 3 cb (G; R) = 0, while Hartnick and the author [28] showed that H 4 cb (G; R) = 0. In large degree, on the other hand, our Theorem 1