Motion algorithms are foundational for effective autonomous robot movement. For surface robotics, one particularly useful algorithm is known as pure pursuit, where a robot follows a point along a path that is a constant distance away from the robot. This work hopes to improve the pure pursuit motion algorithm to account for differences in the robot's features by implementing closed loop full state feedback (FSF) control. In addition, this project aims to provide more abilities to the pure pursuit algorithm, such as specification of angle at each point, allow for moving points, and ensure fast and efficient movements. The algorithm additions are made by modifying the calculations or control loop, and using simulations to verify effectiveness. So far, this work has shown promise by enabling intricate movements while being effective. As a whole, the role of this research is to make pure pursuit more useful and effective for any robot operating on a surface.

Mentored by Kirill Golubnichiy, this research project aims to apply mathematical finance and machine learning (ML) to forecast stock option prices. We create and evaluate new empirical mathematical models for the Black-Scholes equation to analyze data for 177,000 companies. For each company, we have 13 elements including stock and options’ daily prices, volatility, minimizer, etc. Because the market is so complicated that there exists no perfect model, we apply ML to train algorithms to make the best prediction. We first analyze several existing stock and option prediction models: Quasi-Reversibility Method (QRM), Binary Classification, and Regression Neural Network (RNN) ML. QRM is an analytical and analytical approach to find the minimizer by solving the Black-Scholes equation as an ill-posed problem; whereas the latter two combine QRM with ML. The current stage of research attempts to combine QRM with Convolutional Neural Networks (CNN), which learns information across a large number of data points simultaneously. Our current focus is to apply CNN to generate new results by programming, implementing, testing, and validating sample data. We will compare our CNN model with previous models to see if it is possible to achieve a higher profitable rate. If our CNN model successfully forecasts prices for a majority of stock options, it might be possible to deploy the model in the real world and help investors make better investment decisions.

Take a circular sheet of paper. Pick two random points on the edge of the circle and draw a line segment between them. Repeat this many times. Then you take scissors and cut the circle along those line segments. How many pieces of paper do you get? Such questions are a part of a subject called geometric probability. Examples include the famous 18th century Buffon's needle problem that lets one estimate the value of pi by running a random experiment on the computer. In modern times geometric probability is used to estimate the coverage of cell phone towers and communicating drones. I review historical research in geometric probability and graph theory dating between the 18th and 20th centuries to find the expectation and the variance of the number of pieces in the circular sheet. Then I prove a Central Limit-type behavior (converges to normal distribution) of the number of pieces that is conjectured from experiments. The proof is from a modern result of probability theory, called Stein's method for the Central Limit Theorem. Several open questions remain to be answered, including how likely one is to get a piece with a very large or small area. The results of this research enrich the content of geometric probability with possible applications in geology, image processing, transportation, and so on.

When conducting surveys containing sensitive questions, Social Desirability Bias (SDB), people’s tendency to provide socially acceptable answers rather than truthful ones, often leads to low response rate or worse, untruthful responding. Randomized Response Techniques (RRT) combat SDB by allowing respondents to provide scrambled responses. However, if a respondent does not trust the RRT model, data accuracy will be compromised. Lack of trust in binary RRT models has been shown to lead to untruthfulness, and thus unreliable data and unreliable estimates of the sensitive trait. Yet, no quantitative RRT model currently accounts for respondent lack of trust. We propose an Optional Enhanced Trust (OET) RRT model that extends Warner’s Additive Model, a quantitative RRT model with additive noise, by allowing additional multiplicative noise if the respondent does not trust the Warner Additive Model. Using the Unified Measure, a combined metric of respondent privacy and model efficiency, we demonstrate both theoretically and empirically that the proposed OET model is superior to existing models in terms of respondent privacy and estimator accuracy.

When trying to develop the combinatorics of the p-parts of a multiple Dirichlet series given a group of functional equations isomorphic to the Weyl group of a type A root system, one runs into two natural combinatorial defintions of these p-parts in terms of Gelfand-Tsetlin patterns. The fact that these two natural definitions are equivalent is proved in Brubaker, Bump, and Friedberg's Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory. Their proof is not bijective. Further developing the combinatorics of these generalizations of the Riemann zeta function and other Dirichlet series, and finding a bijective proof of the result of Brubaker-Bump-Friedberg, remains an active area of research. I worked with a large group of undergraduate mathematicians, mentored by Ben Brubaker himself as part of the online collaborative Polymath Jr. Program, and we introduced new combinatorial objects, namely a new kind of colored lattice model, with which we can explicitly conjecture the existence of such a weight-preserving bijection for result of Brubaker-Bump-Friedberg in the most general setting, with p-parts corresponding to certain metaplectic Whittaker functions.