Real Algebraic Geometry

on YouTube ↗01h 15m

" summary

Real solutions to polynomial equalities behave non-generic: a univariate quadratic has 0 or 2 real roots, two plane cubics can meet in up to 9 real points and lose them in conjugate pairs, and projecting varieties produces semialgebraic images (example: the polynomial map to 2×2 symmetric matrices yields exactly the PSD cone). Detecting parameter changes uses discriminant chambers found by elimination (Macaulay2 demo). Emptiness of a real variety admits a sum of squares certificate (-1 = SOS + ideal element); testing SOS reduces to a semidefinite program via a PSD Gram matrix, and that relaxation recovers Lovász-style bounds that are tight on perfect graphs.

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" chapters (12)

1. 0:00Introduction to Real Geometry
2. 4:33Challenges of Real Numbers
3. 9:34Projecting Algebraic Sets
4. 15:03Discriminants and Singularities
5. 20:05Analyzing Intersection Points
6. 25:11Topology of Real Curves
7. 29:33Computing Singularity Locales
8. 40:08Certificates of Emptiness
9. 46:54Sums of Squares Algorithms
10. 53:12SDP for SOS Characterization
11. 1:00:10Optimization Applications
12. 1:06:56Q&A and Concluding Remarks