I completed my Master’s Thesis at Cal Poly SLO under the direction of Dr. Ryan Tully-Doyle. You can download to the full document here.
Free Analysis seeks to understand functions that accept matrices of arbitrary sizes as inputs—so called “free” functions. For example, polynomials in a single variable have a natural extension to matrices. Polynomials in several variables are more complicate to extend to matrices, but such extensions are possible and explored at length.
Many classical results (monodromy, Oka-Weil, the implicit function theorem, etc.) have been extended to this new context. At the time of writing, there is no canonical topology for these spaces of matrices (although there are several candidate topologies). My thesis explored one of these topologies in depth, working through and expanding upon the results of J. E. Pascoe in this paper.
My thesis uses tools from a number of different disciples (algebraic topology, algebraic geometry, complex analysis, and even a handful of combinatorial arguments) in order to understand the “shape” of nice1 open subsets of \(\mathcal{D} \subset \bigcup_{\mathbb{N}} M_n(\mathbb{C})\) (called free sets). There are three types of functions defined on such a \(\mathcal{D}\): free functions (which act like classical analytic function), determinental functions (which act like \(\det\)) and tracial functions (which act like the trace).
Directly analogous to the construction in Complex Analysis, we use the analytic continuation of frees function (and the related functions, which can also be analytically continued) defined on \(\mathcal{D}\) create analogues of the fundamental group and detect its topology. Since there is more than one class of function which can be analytically continued, there turn out to be three such fundamental groups: \(\pi_1^{free},\pi_1^{det}, \text{ and }\pi_1^{tr}\). It turns out that \(\pi_1^{free}\) is trivial and \(\pi_1^{tr}=\pi_1^{det}\) is countable, torsion free, and abelian.
It turn out that free and tracial functions defined on some free sets are related by a derivative-like object. As such, Pascoe uses this gradient to construct a basic2 cohomlogy theory analogous to De-Rham cohomology. This cohomology theory allows us to actually compute \(\pi_{1}^{tr}\) for simple spaces.
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See the thesis for details, but in short “niceness” requires the sets are closed under direct sum (\(X \in \mathcal{D} \implies X \oplus X \in \mathcal{D}\)) and under similarity (\(X \in \mathcal{D} \implies U X U^{*} \in \mathcal{D}\) for all like-size unitary \(U\)). ↩︎
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Basic meaning that only \(H^1\) is defined. ↩︎