Prior experience with mathematical software (e.g. Matlab, Maple, Mathematica or Magma). Willingness to get up early in the morning.

**Syllabus:** Convexity,
Semidefinite Programming, Linear Matrix Inequalities, Polynomial
Optimization, Computational Algebra, Real Algebraic Geometry.

** Objective:**
This class offers an
opportunity for graduate students with different backgrounds
and interests to learn from each other.

To accomplish this, participants will work
on course projects, ideally in teams of two or three,
and present their findings in class.

**Reading:** The material will be drawn from various
articles and books. The following titles are
on 1-day reserve in the Math Library:

A. Barvinok: A Course in Convexity, American Math. Soc., 2002, QA639.5 .B37 2002

S. Basu, R. Pollack and M-F. Roy: Algorithms in Real Algebraic Geometry, Springer, 2006, QA564 .B38 2006

S. Boyd and L. Vandenberghe:
Convex Optimization, Cambridge University Press, 2004,
QA402.5 .B69 2004

D. Cox, J. Little and D. O'Shea: Using Algebraic Geometry, Springer, 1998, QA564 .C6883 1998

J. Lasserre: Moments, Positive Polynomials and their Applications,
Imperial College Press, London, 2010, QA402.5 .L377 2010

M. Marshall: Positive Polynomials and Sums of Squares, American Math. Soc, 2008,
QA3 .M283 no.146

Pablo Parrilo's course notes
Algebraic Techniques and Semidefinite Optimization
from Spring 2010 at MIT.

My book
Solving Systems of Polynomial Equations,
Amer.Math.Soc., CBMS Regional Conferences Series, No 97, Providence, 2002.

**Software:** Students are encouraged to familiarize
themselves with some software tools. Possibilities include

Bertini,
CVX,
GloptiPoly, Macaulay2,
QEPCAD,
SAGE,
Singular,
SOSTools,
Surfex,
YALMIP.

**Schedule of lectures:**

August 26: Introduction to Spectrahedra

August 31:
Minimizing Polynomial Functions

September 2: Convex Hull of a Space Curve

September 7:
Philipp Rostalski:
Wiki and Numerical Software

September 9: Polytopes and Linear Programming

September 14: The Central Path

September 16:
Cynthia Vinzant:
The Real Nullstellensatz

September 21: Convex Bodies and Their Algebraic Boundary

September 23: Raman Sanyal:
Orbitopes and Theta Bodies

September 28: Lagrange Duality and Projective Duality

September 30:
Shaowei Lin: Symbolic Software

October 5: The Optimal Value Function

October 7: Petter Branden:
Hyperbolic Programming

October 12: Spectrahedra and Semidefinite Programming

October 14: Quartic Curves and Their Bitangents

October 19: Spectrahedral Shadows

October 21:
Angelica Cueto and
Felipe Rincon: Discriminants

October 26: Nonnegative Polynomials and Sums of Squares (after Greg Blekherman)

October 28: Nonnegative Polynomials and Sums of Squares (after Greg Blekherman)

November 2: Daniel Plaumann: Computing Linear Matrix Representations

November 4:
Ngoc Tran and
Volkmar Welker:
Mathematics of Statistical Ranking

November 9: Patrik Noren and Cynthia Vinzant: Convex Hulls of Monomial Curves

November 11: No class: Administrative Holiday

November 16: Avinash Bhardwaj and Anand Kulkarni: Recognition and Types of
Spectrahedra

November 18: No class: Please attend the
MSRI workshop

November 23: No class: Thanksgiving

November 25: No class: Thanksgiving

November 30: Natth Bejraburnin and Jonathan Terhorst

December 2: Jose Rodriguez and Charles Chen

December 6 (Monday, 9:00-noon, 939 Evans): Olya Mandelshtam,
Andreas Gross, Thanh Vu and Qi Zhang

**Homework:**
In the first eight weeks there will
be regular assignments, posted here in pdf format:

Homework 1 is due Tuesday, September 7.

Homework 2 is due Tuesday, September 21.

Homework 3 is due Tuesday, October 5.

Homework 4 is due Tuesday, October 19.

** Course project deadlines:**
The following dates are all Thursdays:

October 21: Project proposal is due

November 18: Preliminary report is due

December 9: Final paper is due

**Grading:**
The course grade will be based on both the homework
and the course projects. No need to worry about this.