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I am currently using PeakDetect for a project, but I'm not sure how the sharpness parameter works. I can qualitatively tell what it's doing, but is there a more analytical/mathematical way to define sharpness?

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We can see from my reimplementation of FindPeaks here that the sharpness parameter is a threshold that discards all peaks for which the negative of the second derivative is smaller than that threshold.

I implemented it like this:

findPeaks1[data_, sigma_, s_] := Module[{sharpness},
  sharpness = ListConvolve[{1, -2, 1}, data];
  Unitize[findPeaks1[data, sigma] + 1 - Unitize[sharpness, s]]
  ]

You can see here the exact definition, how the second derivative is computed and how the threshold is applied.

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