Research Projects
Research Projects
For my dissertation, I studied Applied Mathematics and focused on modeling complex social phenomena. This area of research allowed me to focus on applying mathematics to interdisciplinary fields and identify connections from one subject to another. When I explore the possibility of new projects, I emphasize three questions:
Can I use my mathematical tools to explain the questions that are being asked?
Am I able to learn and incorporate new mathematical tools to explain the questions that are being asked?
Will someone outside of my field have an understanding of what this project will be?
The last question is of utmost importance to me as I am someone who loves to discuss applications of mathematics in different areas. The main reason is that while many people believe mathematics to be a complex and abstract field, mathematics can also be applicable and tangible to the right audience.
Over the years, I have been passionately engaging my students in participating in mathematics research. Through exploration of different topics and understanding what my students are capable of from their courses, I have been exploring projects that can be applicable and attainable for my students. This has been a primary focus of my research.
Topics that I have explored are economics, international relations, Fibonacci sequences, and mathematical modeling.
Below are summaries of my current and past projects.
Wodarz, M., Ma, T., Komarova, N. (2024). "The dependence of K-12 student performance on the household income in U.S. school districts." Submitted for publication.
This study was inspired by Maria Wodarz and their examination of the dependence of K-12 student performance on the household income in California school districts. This study examines the relationship between K-12 student performance and median household income across U.S. school districts, analyzing data from 42 states for English Language Arts (ELA) and Mathematics in 3rd and 8th grades during the 2018-2019 school year. Performance scores, defined as the percentage of students meeting or exceeding grade-level expectations, were correlated with district income. Different mathematical functions were explored to characterize this relationship, showing consistent positive correlations across states. Results reveal that less affluent states exhibit a steeper increase in performance with income compared to wealthier states. Additionally, grade-level and subject comparisons highlight disparities, including a pronounced decline in math performance from 3rd to 8th grade in most districts. These findings underscore the influence of socioeconomic factors on educational outcomes and the variations between subjects and grade levels.
Ma, T., Fleischman, A., Komarova, N., Wodarz, D. (2024-2027). "Collaborative Research: Using mathematics to bridge between evolutionary dynamics in the hematopoietic systems of mice and humans: from in vivo to epidemiological scales."
This project is funded by NSF (grant number DMS-2424854). The human blood contains different cell types that are continuously produced, while older cells die. As this process continues as the organism ages, mistakes are made during cell production, generating mutant cells. These mutants can linger in the blood and become more abundant over time. They can contribute to chronic health conditions and there is a chance that they initiate cancer. It is not well understood why these mutant cells persist and expand. One problem that has held back progress is that for obvious reasons it is impossible to perform experiments with human subjects to investigate this. Mathematics combined with epidemiological data, however, offers a way around this limitation. This project develops mathematical models describing the evolution of mutant cells in the blood over time, using experimental mouse data to define the model structure. New mathematical approaches are then used to adapt this model to the human blood system, by bridging between mathematical models of mutant evolution in the blood, and the epidemiological age-incidence of mutants in the human population. There is a broad public health impact since this work can suggest ways to reduce the mutant cells in patients, which can alleviate chronic health conditions and reduce cancer risk. From the educational perspective, we collaborate with Xavier Unversity of Louisiana, an undergraduate historically black university, to foster enthusiasm in continued education and careers in STEM, and to equip students with knowledge and skills to potentially continue in graduate programs at top universities, thus promoting social mobility.
Ma, T., Vernon, R., & Arora, G. (2024). “Extension on the Generalization of the 2-Fibonacci Sequences.” In preparation.
In 2023, I explored a Number Theory topic with my colleagues at XULA. We explored the generalization of the four different 2-Fibonacci sequences defined by Atanassov. We are currently working on an extension of our paper with current undergraduate students at XULA.
Rippy, H., Ma, T., McKinley, S., "Income Inequality and the mobility between quintiles." In preparation.
To expand my mathematical horizon, I started collaborating with Dr. Scott McKinley of Tulane University in their field of economics. We are currently working with undergraduates at XULA (Haylee Rippy) and Tulane to study the inequality of income and net worth. My contributions to the project are providing mathematical insight to Dr. McKinley's economic insight in deriving a stochastic model to depict the nuances of how an individual moves between quintiles within income brackets. This project is currently in preparation for submission and presentation at a nearby conference.
Teter, A., Burnette, C., & Ma, T. (2022). “Bee Colony Optimization for Traveling Salesperson Problem: Finding optimal tour route to explore New Orleans.” In review.
To continue my effort in exposing undergraduates to research, I am working with Austin Teter (class of 2025) and Dr. Charles Burnette on a project involving mathematical modeling. Tourism is an integral part of the economy of New Orleans. Touring companies, in particular, provide tourists of this wonderful city with enjoyment and stimulate the economy by exploring restaurants, historical landmarks, and other points of interest. However, these companies rely on having their routes optimized to achieve these goals in a timely and efficient manner. Using Google Maps to compute point-to-point distances, various preexisting tour routes are evaluated as a Traveling Salesperson Problem (TSP) to be better optimized using Bee Colony Optimization (BCO) algorithms. This investigation is to utilize a BCO algorithm that solves the TSP to find an optimal path for commercial touring by bus or by walking. My contributions to this project are being the lead mentor for Austin Teter, programming the simulations, and writing various drafts of the paper. This paper is currently under review in the PUMP Journal of Undergraduate Research.
Ma, T., Fernando, A., & Gurski, K. (2024). “A Research Introduction for Early College Students: Based on the OSU-HBCU Pilot Project.” PRIMUS, 34(9), 969-987. https://www.tandfonline.com/doi/full/10.1080/10511970.2024.2396497
As a coordinated effort to inspire HBCU students to pursue graduate studies in Mathematical Biology, Ohio State University (Dr. Janet Best) collaborated with various HBCUs to mentor students through a pilot program. This project aims to engage faculty and undergraduate students in research by scheduling regular early college-level mathematics research presentations with pre- and post-lecture exercises. While these topics were predominantly in the mathematical biosciences area, the concept can be extended to other areas of mathematical research. Three of the six presentations are discussed in this article. The hope is the audience may use these, as presented, for study with students, with minimal start-up investment and to use this as a model for future efforts. My contributions to the project are organizing the details of the talks to discuss in the paper, and writing various drafts of the paper. This paper was published in the journal PRIMUS in Fall 2024.
Ma, T., Vernon, R., & Arora, G. (2023), “Generalization of the 2-Fibonacci Sequences and their Binet formula.” Notes on Number Theory and Discrete Mathematics, 30(1), 67-80, DOI: 10.7546/nntdm.2024.30.1.67-80.
As a way to persistently strive towards expanding my fields of interest, I explored a Number Theory topic with my colleagues at XULA. We are exploring the generalization of the four different 2-Fibonacci sequences defined by Atanassov. In particular, we define recurrence relations to generate each part of a 2-Fibonacci sequence, discuss the generating function and the Binet formula of each of these sequences, and provide the necessary and sufficient conditions to obtain each type of Binet formula. My contributions to the project are providing the mathematical calculations of the Binet Formula for each case, and writing various drafts of the paper. This paper was published in February 2024 in the journal Notes on Number Theory and Discrete Mathematics.
Dong, T., Zheng, R., Ma, T. (2019), "Diplomatic Dynamics of International Treaty Negotiations.” The PUMP Journal of Undergraduate Research, 2, 199-224.
This project was a product of a Senior Capstone course at Dartmouth College. The course itself was a culminating experience for graduating Mathematics majors that involved exploration of mathematical research. Two students, Tianhang Dong and Robin Zeng, proposed an interesting topic about diplomatic relations and how to model treaty negotiations. In the paper, we aim to answer how to quantify the influential factors through the negotiation process. We do so by categorizing countries into different interest groups based on their attitudes towards the treaty and their socioeconomic statuses and modeling the movement of countries between these interest groups. This model is based on an epidemiological approach to quantifying influence. Through our research, we discovered two equilibrium points, which reveal the conditions under which all countries would either support or oppose a treaty. We also ran simulations under several hypothetical scenarios under which equilibria would occur, demonstrating the practical applications of our model. We further analyze the model's real-life application and the influence of certain parameters through two case studies: the first is on the Basel Convention's negotiation process and the second is on the spread of carbon pricing. I mentored and guided my students to create a publishable article for the PUMP (Preparing Undergraduates through Mentoring toward PhDs) Journal of Undergraduate Research. The paper was published in 2019.
Ma, T., & Komarova, N. L. (2019), “Object-Label-Order Effect When Learning From an Inconsistent Source.” Cogn Sci, 43: e12737.
This project was the second chapter of my thesis with my advisor Natalia Komarova. In this study, we investigate how learners handle learning from an inconsistent source of information. We focus on the setting where a learner receives and processes a sequence of utterances to master associations between objects and their labels, where the source is inconsistent by design: It uses both “correct” and “incorrect” object-label pairings. We hypothesize that depending on the order of presentation, the result of the learning may be different. To this end, we consider two types of symbolic learning procedures: the Object-Label (OL) and the Label-Object (LO) process. In the OL process, the learner is first exposed to the object, and then the label. In the LO process, this order is reversed. We perform experiments with human subjects and also construct a computational model that is based on a nonlinear stochastic reinforcement learning algorithm. It is observed experimentally that OL learners are generally better at processing inconsistent input compared to LO learners. We show that the patterns observed in the learning experiments can be reproduced in the simulations if the model includes (a) an ability to regularize the input (and also to do the opposite, i.e., undermatch) and (b) an ability to take account of implicit negative evidence (i.e., interactions among different objects/labels). The model suggests that while both types of learners similarly utilize implicit negative evidence, there is a difference in regularization patterns: OL learners regularize the input, whereas LO learners undermatch. As a result, OL learners can form a more consistent system of image-utterance associations, despite the ambiguous learning task. The paper was published in 2019 in Cognitive Science.
Ma, T., Wood, K., Xu, D., Guidotti, P., Pantano, A., & Komarova, N.L., (2017), “Diversity and Admission Predictors for Mathematics PhD Success.” Notices of the AMS, vol 65, No 6, p 676.
To understand the successes of a PhD candidate, our group collected and analyzed admission data of students from different groups entering a math PhD program at a Southern California university. We observe that some factors correlate with success in the PhD program (defined as obtaining a PhD degree within a time limit). According to our analysis, GRE scores correlate with success, but interestingly, the verbal part of the GRE score has a higher predictive power compared to the quantitative part. Further, we observed that undergraduate student GPA does not correlate with success (there is even a slight negative slope in the relationship between GPA and the probability of success). Finally, a gender gap is observed in the probability of success with female students having a lower probability of finishing with a PhD despite the same undergraduate performance, compared to males. This gap is reversed if we only consider foreign graduate students. We hope that this study will encourage other universities to perform similar analyses, to design better admission and retention strategies for Math PhD programs. A letter to the editor was written regarding this study to the Notices of the AMS. A PDF copy of our report can be found https://arxiv.org/pdf/1803.00595.pdf.
Ma, T., & Komarova, N.L., (2017), “Mathematical Modeling of Learning from an Inconsistent Source: A Nonlinear Approach.” Bulletin of mathematical biology, 79(3), pp.635-661.
My first paper published was a continuation and extension of previous work by (Mandelstham and Komarova). They started the discussion on how children can modify and regularize linguistic inputs from adults. In this paper, we present a new interpretation of existing algorithms to model and investigate the process of a learner learning from an inconsistent source. On the basis of this approach is a (possibly nonlinear) function (the update function) that relates the current state of the learner with an increment that it receives upon processing the source’s input, in a sequence of updates. The model can be considered a nonlinear generalization of the classic Bush–Mosteller algorithm. Our model allows us to analyze and present a theoretical explanation of a frequency-boosting property, whereby the learner surpasses the fluency of the source by increasing the frequency of the most common input. We derive analytical expressions for the frequency of the learner and also identify a class of update functions that exhibit frequency boosting. This paper was published in 2017.