Research Areas

My research focuses on applied mathematics and data science, with emphasis on developing efficient algorithms and theoretical frameworks for real-world problems. Below is an overview of my main research areas. For a comprehensive list of publications, please see my Papers page or Google Scholar profile.

Mathematics of AI and Learning Theory

I work on theoretical understanding of modern machine learning and artificial intelligence. My research addresses fundamental questions about why neural networks and other learning algorithms succeed in practice, including the phenomenon of benign overfitting where models can fit data perfectly yet still generalize well. This work bridges classical statistical learning theory and modern deep learning, providing rigorous mathematical analysis of neural network training dynamics, generalization bounds, and implicit regularization. Understanding these phenomena is critical for developing more robust and reliable AI systems.

Stochastic Optimization and Iterative Methods

I develop and analyze stochastic iterative methods for solving large-scale optimization problems and linear systems. This includes work on randomized Kaczmarz methods, stochastic gradient descent variants, and accelerated methods for non-convex problems. My research explores how randomization can improve convergence rates and scalability, with applications to high-dimensional data analysis, federated learning, and distributed computation.

Machine Learning and Fairness

My research in machine learning emphasizes transparency, interpretability, and fairness. I work on methods including non-negative matrix factorization and topic modeling that provide interpretable results while maintaining strong predictive performance. Recent work addresses fairness in machine learning, exploring how to build algorithms that provide equitable performance across different population groups. I'm also interested in theoretical foundations of learning and robust reconstruction from incomplete data.

Tensor Analysis and Factorization

Tensors (higher-dimensional arrays) appear naturally in many applications including imaging, genomics, and text analysis. My research develops algorithms for tensor decomposition, low-rank tensor recovery, and tensor-based regression. This includes extending sparse recovery and matrix factorization techniques to the tensor setting, as well as developing randomized methods for efficient tensor computations.

Applications with Community Partners

I'm committed to translating mathematical research into solutions for real-world problems. Recent collaborations include work with the Innocence Center on exoneree support and justice initiatives, applying data analysis techniques to improve outcomes for wrongfully convicted individuals. Additionally, I work with LymeDisease.org on statistical analysis and modeling of Lyme disease data to inform patient advocacy, research directions, and public health policy. These partnerships demonstrate the power of collaborative mathematical research in addressing urgent societal challenges.

Additional Interests

Recent work explores connections between numerical methods and machine learning, manifold learning, phase retrieval problems, and applications to medical data analysis. I'm particularly interested in translating theoretical advances into practical algorithms that solve real-world problems with tangible impact.

For the General Public

Artificial intelligence can seem like magic: a program that writes, sees, and predicts. Underneath, it is a huge collection of mathematical puzzles, and many of them are still open. My research is about solving those puzzles, so that the tools shaping our lives are not just powerful but understood.

One thread of my work is about finding answers efficiently. Many problems, from reconstructing an image to training an AI model, come down to solving a vast set of interlocking equations. I develop methods that get there by making many small, randomized corrections, a bit like tuning a radio by nudging the dial one step at a time, instead of trying to solve everything at once. These methods can handle problems far too large for traditional approaches.

Another thread asks why modern AI works as well as it does. Today's most capable systems can learn so thoroughly from their training examples that they seem to have memorized them, yet they still perform remarkably well on new situations. That should not be possible by the old rules of thumb, and it is one of the most interesting open questions in the field. Answering it helps us know when these systems can be relied on. I also design methods that make a system's reasoning easier to follow, and that help check how it treats different groups of people.

Mathematics is also a way of working with people. Through collaborations with community partners, including The Innocence Center, which works to free people who were wrongly convicted, and LymeDisease.org, which supports patients and families living with Lyme disease, I have seen how careful analysis can help real people make better decisions. My hope is that this research makes AI a tool people can understand and trust.