abstract: Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout. / Dissertation/Thesis / Masters Thesis Mathematics 2016

<p>We consider a novel setting for local harmonic analysis on reductive groups motivated by Langlands functoriality conjecture. To this end, we characterize certain non-linear Schwartz spaces on tori and reductive groups in spectral terms, and develop some of their structure in the unramified case, and we derive estimates of their moderate growth at infinity. We also consider non-linear Fourier transforms, and calculate their action on tame supercuspidal representations of $GL_2(F)$ in terms of inducing cuspidal data.</p>

No description available.