A list of topics covered in the course is presented in the calendar.

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


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.


We will use a variety of book chapters and current papers. Some of these are listed in the readings section.


The final grade will be calculated based on the following weights:

Research Project 50%
Homework 25%
Scribe Notes 25%


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.


1 Introduction

Review of Convexity and Linear Programming

2 PSD Matrices

Semidefinite Programming

3 Binary Optimization

Bounds: Goemans-Williamson and Nesterov

Linearly Constrained Problems

4 Review: Groups, Rings, Fields

Polynomials and Ideals

5 Univariate Polynomials

Root Bounds and Sturm Sequences

Counting Real Roots


Sum of Squares

Positive Semidefinite Matrices

Homework 1 out
6 Resultants



The set of Nonnegative Polynomials

7 Hyperbolic Polynomials

SDP Representability

Homework 1 due
8 SDP Representability

Convex Sets in R2 \n \nHyperbolicity and the Lax Conjecture

Relating SDP-representable Sets and Hyperbolic Polynomials


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


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

Zero-dimensional Ideals

Homework 2 due
15 Zero-dimensional Ideals

Hilbert Series

16 Generalizing the Hermite Matrix

Parametric Versions

SOS on Quotients

17 Infeasibility of Real Polynomial Equations


The Zero-dimensional Case


18 Quantifier Elimination


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

Course Info

As Taught In
Spring 2006