**Abstract:** Let *A* ⊆ ℝ*ⁿ* and let 0 < *k* < *n* be a natural number. The *projection problem* of geometric measure theory concerns the set of subspaces *V* ∊ **Gr**(*n*,*k*) such that the orthogonal projection of *A* onto *V* is somehow "smaller" that one would "expect." In particular, we consider the set of subspaces for which Hausdorff dimension of the projection is strictly less than min(dim *A*, *k*). Marstrand's projection theorem provides what is perhaps the most important answer to this question, although the ensuing decades have seen various refinements and generalizations. In this seminar, we discuss the important results of Mattila (1975) and Falconer (1982) bounding the Hausdorff dimension of the exceptional set of projections, and we introduce ongoing research—first due to Orponen (2015)—about when *Hausdorff* dimension can be replaced with *packing* dimension as a measure of the exceptional set.

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# There's always an exception: Packing dimension of exceptional sets of projections

Ryan Bushling, University of Washington

Thursday, April 21, 2022 - 12:30pm to 1:30pm

RAI 116