# Jerusalem Mathematics Colloquium

Thursday, 23rd March 2006, 4:00 pm

Mathematics Building, Lecture Hall 2

#

Victor Vinnikov

(Ben Gurion University)

"Convexity vs Linear Matrix Inequalities:

Commutative and Noncommutative Settings"

## ** Abstract: **
Linear Matrix Inequalities (LMIs) became during the last
decade one of the most popular and useful tools of system
and control theory. This leads naturally to the question:
which convex sets with an algebraic boundary can be
represented by LMIs?

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.

Light refreshments will be served in the faculty lounge at 3:30.

List of talks, 2005-06

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