A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.

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Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres ±Σ(2,3,6k±1). Along the way, we also give an alternative proof of Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.