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:
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.
Each student will also be responsible for editing and/or writing lecture notes from two lectures.
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.
|LEC #||TOPICS||KEY DATES|
Review of Convexity and Linear Programming
Bounds: Goemans-Williamson and Nesterov
Linearly Constrained Problems
Review: Groups, Rings, Fields
Polynomials and Ideals
Root Bounds and Sturm Sequences
Counting Real Roots
Sum of Squares
Positive Semidefinite Matrices
|Homework 1 out|
The set of Nonnegative Polynomials
|Homework 1 due|
Convex Sets in R2 \n \nHyperbolicity and the Lax Conjecture
Relating SDP-representable Sets and Hyperbolic Polynomials
The Bézout and BKK Bounds
Application: Nash Equilibria
Nonegativity and Sums of Squares
Sums of Squares and Semidefinite Programming
Applications and Extensions
Duality and Density
Bridging the Gap
Recovering a Measure from its Moments
A Probabilistic Interpretation
Duality and Complementary Slackness
|Homework 2 out|
Applications and Examples
|Homework 2 due|
Generalizing the Hermite Matrix
SOS on Quotients
Infeasibility of Real Polynomial Equations
The Zero-dimensional Case
Cylindrical Algebraic Decomposition (CAD)
Copositive Matrices and Pólya’s Theorem
Groups and their Representations
|Homework 3 out|
|22||Sums of Squares Programs and Polynomial Inequalities||Homework 3 due three days after Lec #22|