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Abstract: Starships and time machines are staples of science fiction. Yet recently discovered solutions of the Einstein equations of general relativity appear to describe spacetimes with such exotic behaviors-- time travel, wormholes, warp drives. Are such solutions truly physically realistic, or mere mathematical curiosities which cannot be realized by the engineers of even the most "arbitrarily advanced civilization"? These examples will be used to illustrate how applying mathematical physics can illuminate fundamental issues in physics while helping to determine the ultimate bounds on technology.