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We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if$G$is a non-elementary, analytically finite Kleinian group, and its limit set Λ($G$) is connected, then Λ($G$) is either a circle or has dimension strictly bigger than 1.

The theory of virtual fundamental class defines important invariants such as the Gromov–Witten and the Donaldson– Thomas invariants. It has been generalized to the cosection localized virtual cycle which has applications in Seiberg– Witten, Fan–Jarvis–Ruan–Witten and other invariants. In this paper, we prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles.

We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous, provided that the obstacle is Hölder continuous.

We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^ c case. In particular, we prove a K-theory version of the main results of [{\bf H}],[{\bf Liu1}, Theorem B] for a family of almost complex manifolds.