Abstract: The mathematical study of Knot Theory has found practical applications in fields of Physics and Chemistry, while knots have a long history dating back to Celtic artwork and other religious symbols. There are infinitely many ways to represent a single knot, and some knots can be represented in a way that shows their periodic symmetry. This thesis works on an exploration of characteristics of torus and periodic knots, their visual representations, and knot invariants. This begins with torus knots and links of the form (2,q) and develops a recursive formula for knot polynomials of knots with this form.

Abstract: This thesis examines the construction and analysis of the implied volatility surface for equity options using two foundational option-pricing frameworks: the Black–Scholes model and the Binomial Options Pricing Model. Both models theoretically price options based on similar parameters, neither fully captures market dynamics when implied volatility varies across strike prices and maturities. To quantify these deviations, call and put option contracts on two large-cap, technology stocks were collected. Implied volatilities were then extracted by numerically inverting the Black–Scholes formula across a wide range of strike prices and times to expiry. The resulting estimates were used to construct three-dimensional volatility surfaces. The analysis highlights consistent departures from model-assumed constant volatility, including the presence of volatility smiles and skews driven by moneyness and maturity effects. The findings demonstrate that implied volatility surfaces play a crucial role in modern derivatives pricing, hedging, and risk management, and they reinforce the need for dynamic volatility modeling beyond the assumptions of the Black–Scholes and binomial frameworks.

Abstract: Eyes are one of the most sophisticated organs in the human body. Although it can precisely understand visual data, it lacks the ability to track multiple objects simultaneously. To overcome this limitation, we have developed AI models using deep learning, enabling computers to classify images. Specifically, we use a region based convolutional neural network to localize and classify objects within an image, which combines a region proposal algorithm with convolution feature extraction. The model is trained and evaluated on an image dataset from drone footage of football practices, demonstrating how modern computer vision techniques can automate visual understanding. This work highlights the broader potential of deep learning to enhance tasks involving object perception and detection.

Abstract: A partial differential equation (PDE) is an equation involving an unknown function of multiple variables and its partial derivatives. Modern physics is built on conservation laws that quantify physical change over space and time. Because change is expressed mathematically through derivatives, many governing equations take the form of PDEs. However, PDEs are not guaranteed to have closed form analytic solutions. Here we show that a finite difference approach of first and second order central differences enables highly accurate approximations to PDEs that lack such solutions. The majority of real world problems involving PDEs do not have closed form solutions and thus require numerical approximation as a result of irregular boundary conditions, geometric asymmetries, and non linearities. Using test solutions and an analysis of error convergence rates, we demonstrate a thorough approach to numerically solving the wave equation. Ultimately we demonstrate how numerical PDE solvers have cemented their place in experimental physics and engineering, enabling precise predictions when analytic approaches fail.