Georgia Institute of Technology College of Sciences

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, leading to irregular wavy patterns and traveling liquid droplets. Such droplet dynamics are fundamental components in many engineering applications, including mass and heat exchangers for thermal desalination, as well as water vapor and particle capture. Recent experiments demonstrate that critical flow regime transitions can be triggered by varying inlet geometries and external fields. Similar interacting droplet dynamics have also been observed on hydrophobic substrates, arising from interfacial instabilities in volatile liquid films. In this talk, I will describe lubrication and weighted residual models for falling droplets. The coarsening dynamics of condensing droplets will be discussed using a lubrication model. I will also present our recent results on developing optimal boundary control and mean-field control for droplet dynamics. 

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Data imbalance is a fundamental challenge in data analysis, where minority classes account for a small fraction of the training data compared to majority classes. Many existing techniques attempt to compensate for the underrepresentation of minority classes, which are often critical in applications such as rare disease detection and anomaly detection. Notably, in empirical deep learning, the large model size exacerbates the issue. However, despite extensive empirical heuristics, the statistical foundations of these methods remain underdeveloped, which poses an issue to the reliability of these machine learning models.

In this talk, I will examine imbalanced classification problems in high dimensions, focusing on support vector machine (SVMs) and logistic regression. I will introduce a "truncation" phenomenon---which we verifed across single-cell tabular data, image data, and text data---where overfitting in high dimensions distorts the distribution of logits on training data. I will provide a theoretical foundation by characterizing the asymptotic distribution via a variational formulation. This analysis formalizes the intuition that overfitting disproportionately harms minority classes and reveals how margin rebalancing---a widely used deep learning heuristic---mitigates data imbalance. As a consequence, the theory offers both qualitative and quantitative insights into generalization errors and uncertainty measures such as calibration.

This talk is based on a joint work with Jingyang Lyu (3rd-year Stats PhD student) and Kangjie Zhou (Columbia Statistics): arXiv:2502.11323.

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The 2025 Atlanta Science Festival Mathapalooza! will take place in an art gallery at the opening of an exhibition, The Art of Math.  The show will begin with magic by Matt Baker, followed by hands-on art construction, circus acts, "Math Court" skits, and 4-dimensional dance, before ending with a performance by Tracy Woodard of Bach's Crab Canon, with a discussion of its symmetry algebra.  Find out more and get tickets at https://atlantasciencefestival.org/events-2025/1113-mathapalooza-at-the-gallery/