Multidimensional Discrete-Time Systems, Algebraic Curves and Commuting Nonunitary Operators (23rit010)


(Virginia Tech)

Victor Vinnikov (Ben Gurion University of the Negev)


The Banff International Research Station will host the "Multidimensional Discrete-Time Systems, Algebraic Curves and Commuting Nonunitary Operators" workshop in Banff from May 21 to May 28, 2023.

The interplay between unitary colligations / conservative input/state/output systems, realization theory for Schur-class functions, operator model theory, and Lax--Phillips scattering theory has been a major research direction in operator theory, function theory, system theory, and mathematical physics starting in the middle of the last century. During the last several decades this interplay has been generalized to a variety of multidimensional settings. One of the most challenging generalizations has been to the setting of operator model theory for several commuting operators, leading to overdetermined multidimensional systems and involving deep tools from the theory of compact Riemann surfaces and algebraic curves. While considerable progress has been achieved for the case of commuting dissipative operators and continuous-time systems, the case of commuting contractions and discrete-time systems remained so far out of reach. In order to to tackle it, we have to solve two outstanding open problems that we plan to attack during the proposed research in teams at BIRS: the multivariable Halmos dilation problem (embedding commuting contractions into a quasi-unitary commutative operator vessel), and an explicit treatment of the glueing data at the possible singularities of the disciminant curve. Even a partial progress will lead to a major new understanding of one of the central problems of multivariable operator theory --- commuting unitary dilations of commuting contractions, as well as progress in function theory on finite bordered Riemann surfaces (e.g., interpolation problems), multidimensional system theory, and multievolution scattering theory.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).