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Thursday, April 14, 2016 at 4:30 PM in SAS 4201

Josef Schicho, Johannes Kepler UniversityIcosapods and Quartic Spectahedra

Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such $20$-pods, but could not constrain the

legs to have base and platform points with real coordinates.

Spectahedra are the subsets of semidefinite matrices in some given linear space of symmetric matrices. If the matrices are 4x4 and the space has dimension 4 (quartic spectahedra), then the boundary of the spectahedron is contained in a quartic surfaces, called its symmetroid. Generically, the symetroid has 10 isolated double points with complex cooredinates. The type of a generic quartic spectahedron is the pair of integers (a,b) where a is the number of real double points and b is the number of real double points on the boundary of the spectahedron. The possible types have been identfied by Degtyarev and Itenberg in 2010; Ottem, Ranestad, Sturmfels and Vinzant constructed examples of every possibe type.

In this talk, we present an equivalence of the problem of constructing an icosapod with 20 real legs and the problem of contructing a quartic spectahedron of type (10,0).

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Seminar Organizer: C. Vinzant