Prof. Bill Helton
Dept. of Mathematics
University of California San Diego, U.S.A.
Tuesday, 2017-03-28 16:00
IST-Seminar-Room V9.22 - Pfaffenwaldring 9 - Campus Stuttgart-Vaihingen
One of the main developments in optimization over the last 20 years is Semi-Definite
Programming. It treats problems which can be expressed as a linear matrix inequality (LMI). Any
such problem is necessarily convex, so the determining the scope and range of applicability comes
down to the question:
How much more restricted are LMIs than Convex Matrix Inequalities?
There are several main branches of this pursuit. First there are two fundamentally different
classes of linear systems problems. Ones whose statements do depend on the dimension of the system
"explicitly" and ones (called dimension free) whose statements do not. Dimension dependent systems
problems lead to traditional semialgebraic geometry problems, while dimensionless systems problems
lead directly to a new area which might be called noncommutative semialgebraic geometry. The
classic problems of control are dimension free (noncommutative). Indeed, problems which are
entirely specified by a signal flow diagram are dimension free.
In this talk after laying out the distinctions above we give conjectures about the LMI vs
convexity question their current status and pursuits that go beyond this.
Bill Helton typically works on functional analysis problems arising from a variety of areas. His
earlier articles concerned circuit theory, distributed systems, and aspects of the theory of
operators on Hilbert space which come from circuits, systems, differential and integral equations,
spectral theory and non-commutative differential geometry. The theoretical studies of amplifier
design SISO by Youla's and MIMO by Helton were the first papers in H-infinity engineering, and
developed methods used by the originators of H-infinity control. The focus of Helton's recent work
is treating the algebra behind matrix inequalities in a systematic way, computer algebra aimed at
linear systems research where one has unknowns which are matrices. Helton's group is the main
provider to Mathematica of general noncommutative computer algebra capability.
Bill Helton received the bachelor's degree in mathematics from the University of Texas, Austin,
the Master's and Ph.D degree in mathematics from Stanford University. He was at SUNY, Stony Brook,
as an Assistant and Associate Professor. He visited UCLA for six months and subsequently moved to
UC San Diego where he is currently Professor of Mathematics Emeritus and Research Professor. He won
a TAC Outstanding Paper Award and he was a Guggenheim Fellow, is an IEEE Fellow and an AMS