Vortrag im Schwerpunkt Reelle Geometrie und Algebra
Freitag, 10. Juli 2009, um 14:15 Uhr in F426 (Oberseminar) Victor Vinnikov (Be'er Sheva) Noncommutative Convexity and LMIs
The usual question of linear matrix inequality (LMI) representation
of convex sets is as follows: given a convex semialgebraic set
C in R^d, do there exist real symmetric
matrices A_0, A_1,..., A_d such that
The full answer is only known for d=2; at any rate, the set C has to satisfy some fairly restrictive conditions (so called rigid
convexity).
I will discuss the noncommutative analogue of this question, where
we replace C in R^d by a set of d-tuples
of real symmetric matrices of all sizes defined by some noncommutative
polynomial or rational inequalities in d noncommuting variables.
A fascintating fact that has emerged during the last decade or so
in a variety of settings is that the noncommutative situation is often
better behaved than the commutative one. In particular, there is a rough
conjecture that noncommutative convex semialgebraic sets always admit
a noncommutative LMI representation.
After reviewing some of the background, I will present what is perhaps
the most compelling evidence for this conjecture: a theorem of
Bill Helton, Scott McCullough and myself that a sublevel set of a
noncommutative rational function that is regular and convex near the
origin
always admits a LMI representation.
