Monday, October 22, 2018 - 2:30pm to 3:30pm
Event Calendar Category
Other LIDS Events
Speaker Name
Bill Helton
Affiliation
UC San Diego
Building and Room Number
32-D677
Abstract
The talk concerns equations and inequalities for functions having matrix variables. The functions are typically (noncommutative) polynomials or rational functions. Such functions are central to linear systems problems which are specified entirely by a signal flow diagram and $L^2$ performance specs on signals, to free probability (random matrices), and quantum information theory.
Over the last decade a substantial noncommutative (free) real and complex algebraic geometry has developed within the pure mathematics community. For example, last year the behavior of singularities (Nullstellensatz) fell into place. Today's theory operates below the generality seen in linear systems, so implications there are only by analogy. However, one consequence for random matrices gives formulas for the asymptotic eigenvalue densities of a rational function of random matrices.
The talk will focus on the theory, give some high points and describe what is emerging as the most powerful technique in the subject, essentially a multivariable system realization, and describe consequences.
Biography
Bill Helton typically works on functional analysis problems arising from a variety of areas. He was one of the originators of noncommutative geometry. Also 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. The theoretical studies of amplifier design by Youla and by Helton were the first papers in the now ubiquitous area called H-infinity engineering. The focus of Helton’s recent work is treating the algebra behind matrix inequalities in a systematic way; this has necessitated development of real algebraic geometry for non-commutative polynomials. A related interest is computer algebra and Helton’s group is the main provider to Mathematica of general non-commutative 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. He was a Guggenheim Fellow and is an IEEE Fellow and has delivered plenary addresses at conferences ranging from the annual meeting of the AMS, the European Electronic Circuits Society, the Mathematical Theory of Networks and Systems, SIAM Control and Linear Algebra Society meetings.
https://www.math.ucsd.edu/~helton/