* Spectral Theory Sum Rules, Meromorphic Herglotz Functions and Large Deviations*
**Abstract:** After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip-Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip-Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.
## About the speaker
Professor Barry Simon is the (emeritus) IBM Professor of Mathematics and Theoretical Physics at Caltech. He is a leading authority in spectral theory, functional analysis, and quantum mechanics. He has authored more than 400 publications on mathematics and physics. His work has focused on broad areas of mathematical physics and analysis covering quantum field theory, statistical mechanics, Brownian motion, random matrix theory, nonrelativistic quantum mechanics, random and ergodic Schrödinger operators, orthogonal polynomials, and spectral theory.
He received his AB from Harvard in 1966 and his doctorate in physics from Princeton University in 1970 under the supervision of Arthur Strong Wightman. For the next decade, he held a position in mathematics and physics at Princeton University before joining the faculty at Cal Tech as the Sherman Fairchild Distinguished Visiting Scholar in 1980. He joined the faculty permanently at Cal Tech in 1981 and became IBM Professor there in 1984.
Dr. Simon spoke at the International Congress of Mathematics in 1974 and has since given almost every prestigious lecture available in mathematics and physics. He was named a fellow of the American Academy of Arts and Sciences in 2005, and was among the inaugural class of AMS Fellows in 2012. In 2015, Simon was awarded the International János Bolyai Prize of Mathematics by the Hungarian Academy of Sciences. In 2012, he was given the Henri Poincaré Prize by the International Association of Mathematical Physics. The prize is awarded every three years in recognition of outstanding contributions in mathematical physics and accomplishments leading to novel developments in the field.
In 2016, Professor Simon was awarded the prestigious 2016 Leroy Steele Prize for Lifetime Achievement of the American Mathematical Society (AMS) for his "tremendous impact on the education and research of a whole generation of mathematical scientists through his significant research achievements, highly influential books, and mentoring of graduate students and postdocs", according to the prize citation.
Further information on Professor Simon can be found on his Cal Tech website and on his Wikipedia page. For a poster advertising Professor Simon's public lectures, click here. Read More » |