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Given an initial quantum state I> and a final quantum state F> in a Hilbert space, there exist Hamiltonians H under which I> evolves into F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time T? For Hermitian Hamiltonians, T has a nonzero lower bound. However, among nonHermitian PTsymmetric Hamiltonians satisfying the same energy constraint, T can be made arbitrarily small without violating the timeenergy uncertainty principle. This is because the complex path from I> to F> can be made arbitrarily small. The mechanism described here is similar to that in general relativity in which the distance between two spacetime points can be made arbitrarily small if they are connected by a wormhole. This result may have applications in quantum computing. If time permits, the nature of the classical theory that underlies the PTSymmetric quantum theory will be discussed. The solutions to the classical equations of motion have remarkable properties and the particle trajectories can visit multiple sheets of a Riemann surface. In keeping with the mission of the CNLS, nonlinear wave equations that exhibit PT symmetry will also be discussed, the most notable being the Kortewegde Vries equation. Host: Robert Ecke, TCNLS 