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I was very intrigued by a follow-up to the recent numberphile video about divergent series. It was a return to the idea that the sum of the integers greater than zero can be assigned the value -1/12. There were some places this could be used, but as far as I know it was viewed as shaky math by a lot of experts.
As far as I recall the story goes something like this: now, using a new technique Terrence Tao found, a team was seemingly able to "fix" previous infinities in quantum field theory - there's a certain way to make at least some divergent series work out to being a real number, and the presenter proposed that this can be explained as the universe "protecting us" from the infinities inherent in the math.
It made me think about other places infinities show up in modern physics (namely, singularities in general relativity) and whether a technique something like this could "solve" them without a whole new framework like string theory is.