Geometry of hyperbolic polynomials
Final Report Abstract
Hyperbolicity cones are geometric objects that generalize classical polyhedra (such as tetrahedra, cubes, octahedra, etc.) in space, but are not of linear nature. Such a cone is described as the interior part of the solution set of a real algebraic equation in several variables, characterized by a specific hyperbolicity condition, which can be investigated using various methods, such as algebra or convex geometry. Hyperbolicity cones also play a role in various applications. In contrast to polyhedra, which are bounded by a finite number of flat faces, their geometric structure is much more complex and not easy to determine. For instance, we still do not fully understand which geometric entities are possible and what constraints exist. (This problem is only fully solved in dimension 2.) In this project, hyperbolicity cones were studied in greater detail using various new methods, especially from modern algebraic geometry.
Publications
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Kippenhahn's Theorem for Joint Numerical Ranges and Quantum States. SIAM Journal on Applied Algebra and Geometry, 5(1), 86-113.
Plaumann, Daniel; Sinn, Rainer & Weis, Stephan
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Families of faces and the normal cycle of a convex semi-algebraic set. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 64(3), 851-875.
Plaumann, Daniel; Sinn, Rainer & Wesner, Jannik Lennart
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Hyperbolic secant varieties of M-curves. Journal für die reine und angewandte Mathematik (Crelles Journal), 2022(787), 125-162.
Kummer, Mario & Sinn, Rainer
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Plane quartics and heptagons.
D. Agostini, D. Plaumann, R. Sinn & J. L. Wesner
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Adjoints and canonical forms of polypols. Documenta Mathematica, 30(2), 275-346.
Kohn, Kathlén; Piene, Ragni; Ranestad, Kristian; Rydell, Felix; Shapiro, Boris; Sinn, Rainer; Sorea, Miruna-Ştefana & Telen, Simon
