(Ben Gurion University)
"Convexity vs Linear Matrix Inequalities: Commutative and Noncommutative Settings"
It turns out that there are two natural and fundamentally different situations to consider: the commutative situation, where the variables are scalars, and the noncommutative situation, where the variables are matrices appearing in algebraic formulae that respect matrix multiplication.
In the commutative case there are stringent necessary conditions that a convex set has to satisfy in order to admit a LMI representation. These conditions are known to be sufficient in dimension 2 and are conjectured to be sufficient in general. The tools used to attack the sufficiency are tools of classical algebraic geometry.
In the noncommutative case, it is conjectured that every convex set admits a LMI representation. The conjecture has been established for an important special case. The tools come from the newly emerging `noncommutative function theory'.
In the talk I will introduce the problems, present some results and conjectures, and indicate some of the ideas used in proofs, especially for the noncommutative case.
This is a joint work with Bill Helton and Scott McCullough.