Title: Combinatorics and projection theorems
Speaker: Daniel Glasscock (OSU)
Abstract: The image of a "large" subset of the plane under "most" orthogonal projections (to lines through the origin) is "large." Classically, Hausdorff dimension is the measure of "size" and the planar sets are taken to be "fractal;" results go back to Marstrand, Kaufman, and Matilla. By considering finite point sets and cardinality, it's easy to formulate analogous theorems in the discrete setting where incidence geometry is known to give asymptotically correct bounds.
Thomas Wolff wrote in a 1996 survey on the Kakeya problem, "It is sometimes unclear whether applying the combinatorial techniques in the continuum should be simply a matter of extra technicalities or whether new phenomena should be expected to occur." With an emphasis on the combinatorics involved, my talk will address this question: To what extent can the methods and results from incidence geometry be adapted to projection theorems in the continuous setting?