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Venetian blinds, digital sundials, efficient coverings, and Kakeya sets

Alan Chang (Washington University St Louis)
Tuesday, April 2, 2024 - 1:30pm to 3:30pm
PDL C-401
In the first part of the talk, we introduce the Venetian blind construction, a technique in geometric measure theory. One application is the proof of Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. This result has a dual formulation, known as Davies's efficient covering theorem, which states that we can cover any measurable set in the plane by lines without increasing the total measure. 
In the second part of the talk, we discuss the proofs of these classical results as well as some recent results. By using Venetian blinds, we can prove a variant of Davies's efficient covering theorem in which we replace lines with curves. (This is joint work with Alex McDonald and Krystal Taylor.) Another application is the construction of Kakeya sets for curves. (This is joint work with Marianna Csörnyei.)
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