# Lecture Notes

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

Course notes.
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 R2 Hyperbolicity and the Lax Conjecture 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)