The lecture notes provided below are preliminary and an ongoing work.

LEC # | TOPICS | LECTURE NOTES |
---|---|---|

1 |
Introduction
Review of Convexity and Linear Programming |
(PDF) |

2 |
PSD Matrices
Semidefinite Programming |
(PDF) |

3 |
Binary Optimization
Bounds: Goemans-Williamson and Nesterov Linearly Constrained Problems |
(PDF) |

4 |
Review: Groups, Rings, Fields
Polynomials and Ideals |
(PDF) |

5 |
Univariate Polynomials
Root Bounds and Sturm Sequences Counting Real Roots Nonnegativity Sum of Squares Positive Semidefinite Matrices |
(PDF) |

6 |
Resultants
Discriminants Applications The set of Nonnegative Polynomials |
(PDF) |

7 |
Hyperbolic Polynomials
SDP Representability |
(PDF) |

8 |
SDP Representability
Convex Sets in R Relating SDP-representable Sets and Hyperbolic Polynomials Characterization |
(PDF) |

9 |
Binomial Equations
Newton Polytopes The Bézout and BKK Bounds Application: Nash Equilibria |
(PDF) |

10 |
Nonegativity and Sums of Squares
Sums of Squares and Semidefinite Programming Applications and Extensions Multivariate Polynomials Duality and Density |
(PDF) |

11 |
SOS Applications
Moments Bridging the Gap |
(PDF) |

12 |
Recovering a Measure from its Moments
A Probabilistic Interpretation Duality and Complementary Slackness Multivariate Case Density Results |
(PDF) |

13 |
Polynomial Ideals
Algebraic Varieties Quotient Rings Monomial Orderings |
(PDF) |

14 |
Monomial Orderings
Gröbner Bases Applications and Examples Zero-dimensional Ideals |
(PDF) |

15 |
Zero-dimensional Ideals
Hilbert Series |
(PDF) |

16 |
Generalizing the Hermite Matrix
Parametric Versions SOS on Quotients |
(PDF) |

17 |
Infeasibility of Real Polynomial Equations
Certificates The Zero-dimensional Case Optimization |
(PDF) |

18 |
Quantifier Elimination
Tarski-Seidenberg Cylindrical Algebraic Decomposition (CAD) |
(PDF) |

19 |
Certificates
Psatz Revisited Copositive Matrices and Pólya’s Theorem Positive Polynomials |
(PDF) |

20 |
Positive Polynomials
Schmüdgen’s Theorem |
(PDF) |

21 |
Groups and their Representations
Algebra Decomposition |
(PDF) |

22 | Sums of Squares Programs and Polynomial Inequalities | (PDF) |