People often think of Galois theory as a tool for showing the intractability of univariate polynomial equations. This talk presents the opposite perspective: how Galois theory can be used to analyze the unexpected tractability of highly structured systems of multivariate polynomial equations. Numerical monodromy heuristics based on homotopy continuation methods are a key tool, allowing in many cases the analysis of an appropriate Galois group.

In 2017, Vaswani et al. declared “attention is all you need.” What once seemed exaggerated has become increasingly plausible. Transformers now power models across language, vision, and science. But reusing design choices from language models raises a key question: are they optimal for other modalities? This talk examines Transformer-based time-series foundation models, showing how their rank structure, frequency biases, and uncertainty modeling reveal the need for modality-specific design choices.

Abstract: We formulate a momentum space representation of an incommensurate linear Schrödinger equation. We use the representation to find algorithms for a number of fundamental electronic observables such as the local density of states of spatial configurations, total density of states, and the local density of states in momentum space, which is a parallel object to electronic band structure in the absence of periodicity.

Abstract: By climbing the five rungs of Jacob’s ladder, the Density Functional Theory (DFT) hierarchy approaches the “heaven” of chemical accuracy needed for accurate and reliable modeling of molecules and complex materials. In particular, fourth-rung hybrid functionals can provide semi-quantitative accuracy and have therefore been used to generate data for machine-learning (ML) applications for studying important gas-phase systems and reactive processes.

Abstract: Periodic media appear naturally in many areas of physics, and have drawn particular interest especially with the advent of photonic crystals. Understanding how waves propagate in such media is essential for the design of practical applications.

The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension d of the problem increases, the number of iterations required to ensure convergence within a desired error in the W2 metric scales in proportion to d or its square root.