Spectrahedral Representation of Special Hyperbolicity Cones

Project Description

Given the algorithmic importance of semidefinite programming, the last two decades have seen increasing interest in characterizing which convex cones are spectrahedral (that is, feasible sets of semidefinite programming). One particular class of convex cones are hyperbolicity cones (defined by hyperbolic polynomials), which has connections to several areas of mathematics.

An outstanding open question – the generalized Lax conjecture – asks whether every hyperbolicity cone is spectrahedral. In COSW04 and subsequent works, several special classes of hyperbolic polynomials have been studied, and the question of whether such classes of hyperbolicity cones arising from them are spectrahedral remains wide open.

For the reference, see PDF above.

Background Reading

Some useful references for this project are:

• The paper COSW04 and the references therein
• Branden’s papers, in particular this one
• Mario Kummer’s Master’s thesis

Prerequisites

The ideal URA should have a solid command of:

• linear algebra (equivalent of MATH 245, or MATH 235),
• basic abstract algebra (such as the material from both PMATH 336 and PMATH 347),
• complex analysis,
• matroid theory,
• experience writing rigorous proofs

Note: some of the prerequisites can be waived if the student has excellent academic background, such as international olympiads experience, or other similar achievements.

Applying

If you are interested and have the prerequisites for this project, please send me the following by email:

• Transcript
• Resume
• 1-2 paragraphs on why you are interested in this project (in particular, you should understand at least what a hyperbolic polynomial is and what a definite determinantal represenation is). Mario’s thesis is a good place to start.

Note: any applications deviating significantly from the above instructions will be ignored.