The functional calculus allows us to induce a function on matrices with spectrum in some open set from an analytic function on that open set. Similarly, one can consider the tracial functional calculus, where one then takes the trace of the value of the function induced by the functional calculus. Call a germ of an analytic function global if it analytically continues along every path in the ambient domain. One motivation for homotopy, and thus the fundamental group, comes from the theory of analytic continuation. Two paths are homotopic if and only if for every global germ, the continuation along both paths agree. We therefore formulate the notion of a ``fundamental group" in the functional calculus and tracial functnal calculus essentially by considering paths equivalent in terms of global germs, which now need to analytically continue along every path in the space of matrices with spectrum in the domain. Free universal monodromy implies that in the classical functional calculus, the notion is trivial. The tracial fundamental group is, perhaps surprising, abelian and isomorphic to a direct sum of copies of the rationals.

In this talk, we will introduce the functional calculus and homotopy and proceed to the aforementioned results.