A list of topics covered in the course is presented in the calendar.
Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
Besides general mathematical maturity, the minimal suggested requirements for the course are the following: Linear Algebra (e.g., 18.06 / 18.700), a background course on Linear Optimization or Convex Analysis (e.g., 6.251J or 6.255 / 15.093, 6.253), Basic Probability (e.g., 6.041 / 6.431). Familiarity with the basic elements of Modern Algebra (e.g., groups, rings, fields) is encouraged. Knowledge of the essentials of Dynamical Systems and Control (e.g., 6.241) is recommended, but not required.
Bibliography
We will use a variety of book chapters and current papers. Some of these are listed in the readings section.
Grading
The final grade will be calculated based on the following weights:
ACTIVITIES  PERCENTAGES 

Research Project  50% 
Homework  25% 
Scribe Notes  25% 
Homework
Problem sets will be handed out in an approximately biweekly basis and will be due one week later, at the beginning of the lecture on their respective due dates. We expect you to turn in all completed problem sets on time. Late homework will not be accepted, unless there is a prior arrangement with the instructor.
Scribe Work
Each student will also be responsible for editing and/or writing lecture notes from two lectures.
Collaboration Policy
We encourage working together whenever possible: in the tutorials, on the problem sets, and during general discussion of the material and assignments. Keep in mind, however, that the problem set solutions you hand in should reflect your own understanding of the class material. It is not acceptable to copy a solution that somebody else has written.
Calendar
LEC #  TOPICS  KEY DATES 

1 
Introduction
Review of Convexity and Linear Programming 

2 
PSD Matrices
Semidefinite Programming 

3 
Binary Optimization
Bounds: GoemansWilliamson and Nesterov Linearly Constrained Problems 

4 
Review: Groups, Rings, Fields
Polynomials and Ideals 

5 
Univariate Polynomials
Root Bounds and Sturm Sequences Counting Real Roots Nonnegativity Sum of Squares Positive Semidefinite Matrices 
Homework 1 out 
6 
Resultants
Discriminants Applications The set of Nonnegative Polynomials 

7 
Hyperbolic Polynomials
SDP Representability 
Homework 1 due 
8 
SDP Representability
Convex Sets in R^{2 \n \n}Hyperbolicity and the Lax Conjecture Relating SDPrepresentable Sets and Hyperbolic Polynomials Characterization 

9 
Binomial Equations
Newton Polytopes The Bézout and BKK Bounds Application: Nash Equilibria 

10 
Nonegativity and Sums of Squares
Sums of Squares and Semidefinite Programming Applications and Extensions Multivariate Polynomials Duality and Density 

11 
SOS Applications
Moments Bridging the Gap 

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

13 
Polynomial Ideals
Algebraic Varieties Quotient Rings Monomial Orderings 
Homework 2 out 
14 
Monomial Orderings
Gröbner Bases Applications and Examples Zerodimensional Ideals 
Homework 2 due 
15 
Zerodimensional Ideals
Hilbert Series 

16 
Generalizing the Hermite Matrix
Parametric Versions SOS on Quotients 

17 
Infeasibility of Real Polynomial Equations
Certificates The Zerodimensional Case Optimization 

18 
Quantifier Elimination
TarskiSeidenberg Cylindrical Algebraic Decomposition (CAD) 

19 
Certificates
Psatz Revisited Copositive Matrices and Pólya’s Theorem Positive Polynomials 

20 
Positive Polynomials
Schmüdgen’s Theorem 

21 
Groups and their Representations
Algebra Decomposition 
Homework 3 out 
22  Sums of Squares Programs and Polynomial Inequalities  Homework 3 due three days after Lec #22 