Spectrahedra and their Shadows

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" summary

Spectrahedra in three dimensions behave concretely: cubic examples come only in two combinatorial types — samosa vs lemon — with either four or two real nodes and at least one rank‑one boundary point, while quartic symmetroids have ten complex nodes and can realize 20 real (β,σ) patterns; the authors give explicit 4×4 matrix constructions realizing every admissible case (10-node symmetroid, β and σ even, σ≥2). Projection arguments (Cayley/Kummer analysis) produce the plane ramification: a quartic projection yields a degree‑12 curve with 46 singularities, 10 from nodes and 36 from ramification.

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" chapters (20)

1. 0:00Intro
2. 3:33Semidefinite Optimization ... is the computational problem of minimizing
3. 6:25A Familiar Picture The set of 3 x 3 correlation matrices is the elliptope
4. 10:33Sums of Squares
5. 12:42SOS Example
6. 18:18The Quartic Spectrahedron
7. 24:09Sweet Dreams The pillow is a quartic spectrahedron
8. 24:45Symmetroids
9. 29:05Cubic Spectrahedra
10. 32:23Degtyarev-Itenberg Theorem
11. 35:27Twenty Types
12. 36:50Transversal Spectrahedra
13. 37:55Back to Cayley (1869)
14. 45:43Extended Formulations A spectrahedral shadow is a convex set of the form
15. 49:33Ramification
16. 53:04Questions Let S be a generic spectrahedral shadow of type (n.d.).
17. 54:06Algebra
18. 55:56Answers
19. 57:40Numbers
20. 59:23Degeneration A spectrahedral shadow of type (4,2,1) with a parameters