In this paper, for each \(k\in \mathbb{N}\), we define a complex \(\Gamma_k(S)\) for an infinite-type surface \(S\) with non-planar ends, which serves as an analog of the Hatcher-Thurston complex for the infinite-type setting. We show that \(\Gamma_k(S)\) is connected, simply connected, and that the automorphism group of \(\Gamma_k(S)\) is isomorphic to the extended mapping class group.

In a hyperbolic polygon any finite collection of closed billiard trajectories can be assigned an average length function. In this paper, we consider the average length of the collection of cyclically related closed billiard trajectories in even-sided right-angled polygons and the collection of reflectively related closed billiard trajectories in Lambert quadrilaterals with acute angle . We show that in the former case the average length is minimised by the regular evensided right-angled polygon, and in the latter case it is minimised by the Lambert quadrilateral with a reflective symmetry about its long axis. We use techniques from Teichmueller theory to prove the main theorems.

Thurston’s introduction of the asymmetric metric on Teichmuller space brought new interest to the hyperbolic geometric perspective to Teichmuller theory, which earlier was viewed in terms of complex analysis. In this dissertation, I go over several well known results in hyperbolic geometry and Teichmuller theory. Some theorems in Thurston’s seminal paper [Thu98] are proved in this dissertation in detail using the proof ideas presented by Thurston. In the last chapter of this dissertation, the dilation ray construction in [CF21] is shown to be a Thurston geodesic for the case when the geodesic lamination is a pants decomposition.

The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, \(\Phi\), due to a (classical) magnetic field, \({\mathbf B}\), which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, \(\mathbb{R}^2\), of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, \(\mathbb{R}^2-\{0\}\), corresponding to the charge.

We quantize punctured plane, \(X=\mathbb{R}^2-\{0\}\), employing Isham's group theoretic quantization procedure. After sketching out a brief review of group theoretic quantization procedure, we apply the quantization scheme to the phase space, \(M=X \times \mathbb{R}^2\), corresponding to the punctured plane, \(X\). Particularly, we find the canonical Lie group, \mathscr{G}, corresponding to the phase space, \(M=X \times \mathbb{R}^2\), to be \(\mathscr{G} = \mathbb{R}^2 \rtimes (SO(2)\times \mathbb{R}^+)\). We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, \(\mathscr{G} = \mathbb{R}^2 \rtimes (SO(2)\times \mathbb{R}^+)\), and the smooth functions, \(f\in C^{\infty}(M)\), in the phase space, \(M=X \times \mathbb{R}^2\). Making use of this homomorphism and unitary representation of the canonical group, \(\mathscr{G} = \mathbb{R}^2 \rtimes (SO(2)\times \mathbb{R}^+)\), we deduce a quantization map that maps a subspace of classical observables, \(f\in C^{\infty}(M)\), to self-adjoint operators on the Hilbert space, \(\mathscr{H}\), which is the space of all square integrable functions on \(X=\mathbb{R}^2-\{0\}\) with respect to the measure \(\text{d} \mu = \text{d} \phi\text{d}\rho/(2\pi\rho)\).