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Lecture Notes and/or Slides for the

2019 AMS Short Course on

Sum of Squares: Theory and Applications


Each of the six lectures in the course will be broken into two 45-minute sections. Problem sessions are planned for both days, and speakers and organizers will be available to assist participants with these exercises. Some exercises will be facilitated by numerical software such as Macaulay2, Maple™, MATLAB, or Julia.   Participants with a desire to experiment with computations and an interest in applications should find this experience especially stimulating. The speakers will also provide participants with a list of research questions to guide their explorations after the course.

Please bring your laptop to the course, and if possible, download Macauley2 or Julia before the course starts.

This short course is aimed at students with minimal to no background in the theory of SOS polynomials. However, familiarity with properties of polyhedra, convex sets, and polynomials will provide useful background. Appendix A in the book Semidefinite Optimization and Convex Algebraic Geometry, G. Blekherman, P.A. Parrilo and R.R. Thomas (eds.), MOS-SIAM Series on Optimization, SIAM 2012, is recommended as useful preparatory reading. A free pdf of the book can be found at


Please look at this file and download the required code:

Getting Started with Sums of Squares:


Introduction to Sums of Squares (Slides) (updated)


Greg Blekherman, Georgia Tech


Engineering applications of SOS polynomials



Georgina Hall, INSEAD


Lifts of convex sets



Hamza Fawzi, University of Cambridge


Rounding Sum-of-Squares (Lecture 1)

Lower Bounds Against Sum-of-Squares (Lecture 2)


Ankur Moitra, MIT


Algebraic geometry through the lens of sums of squares

Slides Part 1

Slides Part 2


Mauricio Velasco, Universidad de los Andes


Geometry of Spectrahedra (Slides)


Cynthia Vinzant, North Carolina State University





Day 1: 

8:30-9:15   Lecture 1 Greg Blekherman 

9:25-10:10  Lecture 1, continued

10:10-10:40 Coffee Break

10:40-11:25 Lecture 2 Cynthia Vinzant 

11:35-12:20 Lecture 2, continued

12:20-2:00 Lunch

2:00-2:45 Lecture 3 Georgina Hall  

2:55-3:40 Lecture 3, continued

3:40-4:10 Coffee Break

4:10-5:30 Problem session 



Day 2: 


8:30-9:15   Lecture 4  Mauricio Velasco 

9:25-10:10  Lecture 4, continuted

10:10-10:40 Coffee Break

10:40-11:25 Lecture 5 Ankur Moitra 

11:35-12:20 Lecture 5, continued.

12:20-2:00 Lunch

2:00-2:45 Lecture 6  Hamza Fawzi

2:55-3:40 Lecture 6, continued 

3:40-4:10 Coffee 

4:10-5:30 Problem session