My research focuses on applying the skills I've acquired to ensure a thorough understanding of them. Throughout this process, I continuously learn and develop new disciplines. Currently, I am deepening my understanding of applied linear algebra through a combination of research and studying Introduction to Applied Linear Algebra by Stephen Boyd and ​Lieven Vandenberghe. Additionally, I plan to expand my knowledge of computational & numerical methods and functional analysis. To acquire skills outside of my research, I engage in personal projects. For instance, I am developing a climate model using random forest and LSTM techniques to enhance my expertise in machine learning, probability theory, statistical analysis, and environmental and atmospheric data analysis.

Research

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Positive Matrices and Extendability

Along with my colleagues under the supervision of Professor Thomas Sinclair, I'm investigating positive mappings and their extensions. We are working on a paper that has established a thorough groundwork for understanding positive mappings and their lifts via the untanglement mapping, 𝜏. The paper demonstrates the essential properties of 𝜏 and the matrix representations of 𝜏 and its inverse, revealing their influence on positive mappings. Finally, the paper presents and proves the Untanglement Theorem, which asserts that a positive lifting exists if and only if the map can be untangled, thus exploring its implications in lifting positive maps. The code can be found in the GitHub and draws on a blend of disciplines including linear programming, optimization, and machine learning (particularly SVMs). We developed methods to create, classify, and visualize the mappings.

We proved Farkas' lemma using and connecting to the hyperplane separation theorem (proof) using basic linear algebra and convex optimization. We also developed methods to verify classifiers to distinguish between liftable and nonliftable maps. Furthermore, we acquired insights into positive mappings and how to make them extend in our paper.

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Martian Climate Research

During my time at Purdue Earth, Atmospheric, and Planetary Sciences (EAPS), I contributed to a research project on Martian climate alongside Audrey Durham and Benjamin Carpenter, under the guidance of Professor Lei Wang and graduate student Zhaoyu Liu. My research focused on understanding Mars' climate through the study of annular modes using the EMARS and MACDA datasets, drawing comparisons to Earth's atmospheric dynamics. I engaged in extensive literature review, data analysis using netCDF formats and Python, and regular collaborative meetings. Additionally, I prepared and presented my research interests to my lab and submitted an abstract to AGU and participating in various workshops.

One of my key moments of growth was studying the periodic variability in Mars' storm tracks and their relationship with dust events, identifying significant midlatitude eddy activity influenced by the southern hemisphere's topography. I understood the importance of annular modes in Mars' and Titan's atmospheres, revealing parallels with Earth's atmospheric behavior. This was a paper written by Michael Battalio, who I was mentored by for a short time. Participating in the Geophysical Fluid Dynamics (GFD) Bootcamp further enriched my understanding of critical atmospheric concepts like Rossby waves, vorticity, baroclinic instability, and Quasi-geostrophic models. These efforts contribute to my broader goal of advancing planetary climate research and developing a long-term research career in this field.

I read Adam Showman's Atmospheric Circulation of Exoplanets to serve as my introduction to atmospheric dynamics and exoplanetary atmospheres. I also dipped my toes into the works of Wanying Kang's articles and Professor Dennis Hartmann's notes on objective analysis.

Personal Projects

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West Lafayette Climate Model

I am developing a climate model and weather prediction system for West Lafayette, Indiana, using an ensemble model to study, test, and predict its future climate. Currently, I am refining my Random Forest and LSTM models. This project marks my first experience working with a large weather dataset spanning decades, using publicly available data from NOAA's Local Climatological Data. Through this process, I've honed my ability to read, clean, and utilize complex weather data for temperature prediction. Tackling challenges like missing values and outliers required creative solutions; for instance, instead of using a simple average across an entire column for missing temperature values, which could skew results, I applied a moving average within a specific window to maintain accuracy over time. This project has significantly strengthened my familiarity and experience with machine learning techniques, allowing me to deeply understand and fine-tune the process to achieve my desired outcomes.

I intend to use SHAP to explain model predictions and develope an automated data pipeline to continuosly update the model with new data.

Mentors

​Among my many role models, Professors Sam Nariman & Thomas Sinclair are perhaps the most influential. Professor Nariman's mentorship ignited my passion for linear algebra and revealed its profound beauty. Through his guidance in Linear Algebra II, I discovered an affinity for a subject I once found daunting. Professor Sinclair's mentorship throughout my undergraduate research has shaped me into a self-improving mathematician. Lubna of Córdoba, a renowned mathematician and intellectual leader of Andalusia, also inspires me with her excellence in math and stewardship of numerous libraries. I draw further inspiration from László Lempert, Roger Heath-Brown, and Maryam Mirzakhani, whose remarkable achievements and dedication enrich my journey.

Professor Lei Wang is another key influence, motivating me to delve into atmospheric dynamics and exoplanetary research. His guidance throughout my first research experience taught me how to be a researcher and read scientific papers with a keen eye. Moreover, he introduced me to many key figures in the atmospheric dynamics community. I also admire Dr. Karen Smith and Dr. Claire Monteleoni for their pioneering work in environmental data science (EDS). Through their work and lectures, they've introduced me to communities and opportunities within the EDS community.