Introduction to Free Noncommutative Geometry and Analysis

Introduction to Free Noncommutative Geometry and Analysis

The purpose of this minicourse is to present some recent mathematical developments that are concerned with ``free noncommutative analogues'' of well known concepts and results in algebra, geometry, and analysis.

Tentative syllabus (I will assume only the basic background that is usually covered during the first two years of study, and discuss the more advanced notions and results that I will need):

           Background from the theory of free rings. Construction of the free skew field. Realization theory of noncommutative rational functions. (A quick reference is [5], two detailed background treatises are [2] and [7].)

           Background from the theory of operator algebras. Matrix convexity. Noncommutative linear matrix inequalities. (Some references are [3] and [4].)

           Introduction to free noncommutative function theory and to noncommutative differential calculus. Noncommutative analogues of classical interpolation problems. (Some references are [6] and [1].)


[1] J.A. Ball, G. Marx, and V. Vinnikov. Interpolation and transfer-function realization for the noncommutative Schur-Agler class. Operator Theory: Adv. Appl. (to appear), preprint arXiv:1602.00762.
[2] P. M. Cohn. Free ideal rings and localization in general rings. Cambridge University Press, Cambridge, 2006. New Mathematical Monographs 3.
[3] E. G. Effros and Zh.-J. Ruan. Operator spaces. London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York, 2000.
[4] J. W. Helton, I. Klep, and S. McCullough. Free Convex Algebraic Geometry. Semidefinite Optimization and Convex Algebraic Geometry (ed. by G. Blekherman, P. Parrilo, and R. Thomas), pp. 341405, SIAM, 2013. (arXiv version)
[5] D. S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov. Noncommutative rational functions, their difference-differential calculus and realizations. Multidimens. Syst. Signal Process. 23 (2012), no. 12, 4977. (arXiv version)
[6] D.S. Kaliuzhnyi-Verbovetzkyi and V. Vinnikov. Foundations of Noncommutative Function Theory. Math. Surveys and Monographs 199, Amer. Math. Society (2014). (arXiv version)
[7] L. H. Rowen. Polynomial identities in ring theory, volume 84 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.

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