In this thesis, it is proposed to examine the difference equation:(z-h) ∆-hW(z) = A(z)W(z)(1) where W(z) is a vector with two components,∆-hW(h) = W(z) – W(z-h)/h(2)Here, A(z) is a 2x2 matrix, whose elements admit factorial series representations:A (z) = R + Σ∞s=0 As+1S!/z(z+h) ••• (z+sh)(3)R and As+l are square matrices of order two and independent of z. We also assume that eigen values of R do not differ by an integer. We hope to show that if (3) converges in some half plane, then (1) will have a fundamental matris solution of the form: W(z) = S(z)ZR where S(z) is a 2x2 matrix, whose elements have convergent factorial representation in some half plane.

Landau-Ginzburg Mirror Symmetry provides for the construction of two algebraic objects, called the A- and B-models. Special cases of these models–constructed using invertible polynomials and abelian symmetry groups–are well understood. In this thesis, we consider generalizations of the B-model, and specifically address the associativity of the multiplication in these models. We also prove an explicit B-model isomorphism for a class of polynomials in three variables.

In the first part of the dissertation, we construct two-dimensional TQFTs which categorify the evaluations of circles in Kuperberg’s 𝐵₂ spider. We give a purely combinatorial evaluation formula for these TQFTs and show that it is compatible with the trace map on the corresponding commutative Frobenius algebras. Furthermore, we develop a theory of Θ-foams and their combinatorial evaluations to lift the ungraded evaluation of the Θ-web, thus paving a way for categorifying 𝐵₂ webs to 𝐵₂ foams. In the second part of the dissertation, we study the calculus of unoriented 𝔰𝔩₃ webs and foams. We focus on webs with a small number of boundary points. We obtain reducible collections and consider bilinear forms on these collections given by pairings of webs. We give web categories stable under the action of certain endofunctors and derive relations between compositions of these endofunctors.

Quantum information brings together theories of physics and computer science. This synthesis challenges the basic intuitions of both fields. In this thesis, we show that adopting a unified and general language for process theories advances foundations and practical applications of quantum information. Our first set of results analyze quantum algorithms with a process theoretic structure. We contribute new constructions of the Fourier transform and Pontryagin duality in dagger symmetric monoidal categories. We then use this setting to study generalized unitary oracles and give a new quantum blackbox algorithm for the identification of group homomorphisms, solving the GROUPHOMID problem. In the remaining section, we construct a novel model of quantum blackbox algorithms in non-deterministic classical computation. Our second set of results concerns quantum foundations. We complete work begun by Coecke et al., definitively connecting the Mermin non-locality of a process theory with a simple algebraic condition on that theory's phase groups. This result allows us to offer new experimental tests for Mermin non-locality and new protocols for quantum secret sharing. In our final chapter, we exploit the shared process theoretic structure of quantum information and distributional compositional linguistics. We propose a quantum algorithm adapted from Weibe et al. to classify sentences by meaning. The clarity of the process theoretic setting allows us to recover a speedup that is lost in the naive application of the algorithm. The main mathematical tools used in this thesis are group theory (esp. Fourier theory on finite groups), monoidal category theory, and categorical algebra.

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