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].)
References:
[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. 341–405, 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. 1–2, 49–77.
(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.
Please contact me if you have any questions!