The free skew field from (mostly) free noncommutative analysis perspective

The free skew field, introduced by S. Amitsur and studied in depth by P.M. Cohn, is the universal skew field of fractions of the free associative algebra. During the last two decades it came to play an important role  in free noncommutative analysis, the theory of functions of free noncommuting variables represented by matrices of increasing size, that was introduced by J.L. Taylor in his work on noncommutative spectral theory and further developed by D.-V. Voiculescu with a view towards free probability.

In this talk I will introduce the free skew field from free noncommutative analysis perspective. I will discuss the difference-differential calculus and power series expansions (Taylor--Taylor series, named after Brook Taylor of the calculus fame and J.L. Taylor), and their relations to realizations that originated simultaneously in automata theory and in systems and control. Time permitting I will touch on some other results such as the construction of the universal field of fractions of a tensor product of free algebras and the absence of rational identities for free noncommutative random variables (the work of T. Mai, R. Speicher, and S. Yin).

This will be a survey talk, so the content is the work of many authors. My own work on the topic has been done in collaboration with D. Kaliuzhnyi-Verbovtskyi, with I. Klep and Ju. Volcic, and with M. Porat.

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