It is rare for the theory of computing to be used to answer open mathematical questions whose statements do not involve computation or related aspects of logic. This talk discusses recent developments that do exactly this. After a brief review of algorithmic information and dimension, we describe the point-to-set principle (with N. Lutz) and its application to new results in geometric measure theory. These include the following.

1) Strengthening lower bounds on the Hausdorff dimension of generalized Furstenberg sets.  (N. Lutz and D. Stuff, 2017)

2) Extensions of fractal intersection formulas for Hausdorff and packing dimensions in Euclidian spaces from Borel sets to arb. sets. (N. Lutz, 2017)

3) Extensions of Marstrand's projection theorem from anaytic sets to arb. regular sets.  (N. Lutz and D. Stull, 2018)