This talk explores how to construct meaningful features from noisy, high-dimensional data by leveraging geometric and invariant structures. First, we introduce a geometric framework for dimension reduction using a power-weighted path metric, which effectively de-noises high-dimensional data while preserving its intrinsic geometric structure. This framework is particularly useful for analyzing single-cell RNA data and for multi-manifold clustering, and we provide theoretical guarantees for the convergence of the associated graph Laplacian operators.

The Census is our most fundamental tool for measuring who lives in the United States, and where. Ever since the founding of the country, the categories have reflected how Americans think about ourselves, and the data gets used for everything from funding allocations to political districting. So, it might be surprising that the most recent Decennial Census included intentional injections of random numbers to noise the data for privacy protection. This was hugely controversial! In this talk, Moon Duchin will explain some of the history and the impacts of how we enumerate the country.

Nonlinear polynomial equations naturally arise throughout mathematics, science, and engineering with their solutions describing various phenomena including the motion of a mechanical linkage such as a robotic arm and steady states of a dynamical system arising from a biochemical reaction network. Polynomials are central to some of the deepest mathematics and have been studied for millennia.