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.

No abstract uploaded!

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to$S$^{1}-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small$J$-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small$J$-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

No abstract uploaded!

No abstract uploaded!