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Sums of squares in convex algebraic geometry

Subject Area Mathematics
Term from 2014 to 2021
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 253289397
 
Final Report Year 2021

Final Report Abstract

This project was the successor to a previous project, the main result of which had been the disproof of the Helton-Nie conjecture. Some of the main results of the present project concern upper bounds on the complexity of semidefinite representations. In particular, it was proved that every closed convex set in the plane is second-order cone representable. Essentially our proof is even constructive. In joint work with Averkov we tried to extend this result to convex hulls of curves in n-space, the expected result being that they have lifted LMI representations that are block diagonal sums of representations by matrices of size ≤ n+1 . At least for monomial curves this has been shown, and again the approach is constructive. We recon- sidered an important theorem by Helton-Nie, according to which compact convex sets with nonsingular boundary of strict positive curvature are spec- trahedral shadows. We strengthened this result considerably by showing that such sets are even second-order cone representable. On another line, we studied Gram spectrahedra, they parametrize the sums-of-squares repre- sentations of polynomials. Together with two PhD students, we studied the ranks and dimensions of faces and obtained a series of results for quadratic forms on varieties of minimal or almost minimal degree. Generally they concern restrictions on possible dimensions and/or ranks of faces, or other properties of faces like being polyhedral. Finally, in joint work with Kobert we strengthened results from the previous project on polar orbitopes. As one of the consequences, we discovered large classes of previously unknown doubly spectrahedral sets (in fact, all polar orbitopes are of this sort).

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