In the first part of this talk, we introduce the algorithmic techniques that have recently been used in several areas of geometric measure theory. In particular, we will introduce the concepts of Kolmogorov complexity and effective dimension. We will describe the "point-to-set principle", which relates the algorithmic theory with Hausdorff dimension.

In the second part of this talk, we discuss a recent application of the algorithmic techniques to projection theorems in geometric measure theory. Marstrand’s projection theorem states that, for any analytic , for almost every direction , the Hausdorff dimension of the orthogonal projection of onto , , is maximal. A natural question is whether there are non-trivial subsets of directions that work for every (analytic) set . We call such a set of directions universal. Surprisingly, universal sets exist. We prove the existence of universal sets using techniques from computability theory, and, in particular, effective dimension. This is joint work with Jacob Fiedler.