Abstract:
I will discuss two problems; one is very old and the other is new. The first is the "Opaque Square" problem: if we have a union of line segments \$X\$ with the property that every line that intersects the unit square \$[0,1]^2\$ also intersects the set \$X\$, how long does \$X\$ have to be? It has a nice interpretation involving finding buried telephone lines. There hasn't been very much progress since the PhD thesis of Jones (1962). The other problem is recent: if we have a set of points with maximal distance 1 and there are many pairs of points whose distance is close to maximal, then there are also many points clustered together; it should have a simple solution, I will present an easy-ish intermediate result.
Note: The main talk starts at 4:10. There is no pre-seminar.
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Meeting ID: 915 4733 5974