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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

Abstract: Let A ⊆ ℝ 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 Ak). 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.