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Abstract: Julius Plücker introduced coordinates of lines in three-space; they are dual to coordinates of points in the same space. The variety of all these lines is a quadric hypersurface in five-space, the Plücker-Quadric.
Ruled surfaces are one-parameter families of lines, they correspond to curves in the Plücker-Quadric. In this talk we give an outline of the development of these ideas, in particular generalisations to arbitrary dimensions.
Colloquium - 16:00
Abstract: Every convex polygon with integral vertices exactly one integral point in the interior has a dual with the same properties: it is reflexive. Poonen and Villegas give four proofs that their lengths add up to 12. The only proof known (to me) of a similar fact for 3-dimensional reflexive polytopes uses the detour/shortcut via algebraic geometry: K3 surfaces have Euler characteristic 24. There are interpretations of these results as discrete Gauss-Bonnet Theorems.
In this talk I want to use the "12" in order to classify all triples (a,i,p) such that there is a convex polygon with integral vertices which has area a, and p respectively i lattice points on the boundary respectively in the interior. If time permits, I also introduce an onion skin parameter l, and classify quadruples.
No deep math should be expected. The talk should be very self-contained, and, with some luck, entertaining.