Density functional theory is a standard tool for computing energies (eigenvalues) throughout chemistry and in many parts of physics and materials science. I will present the basics of the subject and then show how semiclassical analysis can improve the accuracy of its results. Specifically, we compensate errors in approximate density functionals using a normalization correction derived from semiclassical spectral asymptotics.

Diffusion models are powerful generative systems, yet their internal mechanisms remain difficult to analyze. Taking a physicist's approach, we study the simplest tractable case: a diffusion model with a linear score function.

A key duality links architecture and distribution — a Gaussian dataset implies a linear optimal score, and a linear score network implies the learned distribution is the Gaussian approximation of the data. This duality enables fully analytical treatment across four aspects of diffusion models. Sampling dynamics. The linear score yields a closed-form, low-dimensional, rotation-like sampling trajectory governed by data covariance — and precisely predicts the early phase of pretrained diffusion models, revealing dominant linear structure across a wide range of noise scales. Learning dynamics. Deep linear networks admit analytical training dynamics, uncovering a spectral bias: structure is learned first along the top eigendimensions of the data. Receptive field structure. The effective receptive field is shaped by data covariance rather than architectural priors — it need not be local or equivariant — yielding predictions that extend recent work by Kamb and Ganguli. Sample consistency. Using random matrix theory, we predict sensitivity to dataset resampling, identifying which noise seeds yield consistent versus variable outputs. This work shows how a tractable linear regime provides a rigorous analytical lens into the sampling, learning, receptive field structure, and consistency of diffusion models — with insights that extend surprisingly far into the nonlinear setting.

Many state-of-the-art methods in machine learning are black boxes which do not allow humans to understand how decisions are made. In a number of applications, like medicine and atmospheric science, researchers do not trust such black boxes. Explainable AI can be thought of as attempts to open the black box of neural networks, while interpretable AI focuses on creating clear boxes. Adversarial attacks are small perturbations of data that cause a neural network to misclassify the data or act in other undesirable ways. Such attacks are potentially very dangerous when applied to technology like self-driving cars. The goal of this talk is to introduce mathematicians to problems they can attack using their favorite mathematical tools. The mathematical structure of transformers, the powerhouse behind large language models like ChatGPT, will also be explained.

Two convex bodies $K$ and $L$ in $\mathbb{R}^n$ are affine-equivalent if there exists an affine transformation $T$ such that $T(K)=L$. To measure how close two convex bodies are when they are not exactly affinely equivalent, one introduces the Banach--Mazur distance. Roughly speaking, this is the smallest factor $R \ge 1$ such that, after an appropriate affine transformation, one body is contained in the other, and the other is contained in its dilation by $R$ (with respect to some center $\xi$).

A consequence of Fritz John’s theorem is that for any symmetric convex body, one can approximate $K$ by a polytope $P$ with $O(n)$ facets and Banach--Mazur distance $O(\sqrt{n})$, which is sharp for the Euclidean ball. In contrast, for general convex bodies, the same theorem implies that even with $O(n^2)$ facets, one can only guarantee distance $O(n)$. Thus, there is a gap of order $\sqrt{n}$ for Coarse Polytope Approximation (coarse means the allowance of facets/vertices can only be polynomial in $n$.)

While this problem has been known for over two decades, it remains open whether this $\sqrt{n}$ gap is essential (up to polylogarithmic factors).

In this talk, we will. show that the $O(n)$ bound is essential if one requires the scaling center to be a classical center, such as the barycenter. In other words, either the $\sqrt{n}$ gap is inherent for general convex bodies, or one must go beyond such classical choices of centers. We also include a concrete open problem that appears approachable with current technique is also included.

(This is joint work with Mark Rudelson.)

Abstract: Generative AI has rapidly transformed machine learning, with diffusion and autoregressive models achieving unprecedented performance across vision, language, and scientific discovery. Despite this success, our theoretical understanding still lags far behind practice: why do these models generalize so effectively from finite data in high dimensions? In this talk, I present a mathematical framework that shows that intrinsic low-dimensional structure is the key to understanding this phenomenon and provides a foundation for building more trustworthy generative AI. Through the lens of mixtures of low-rank Gaussian models, I show that learning high-dimensional distributions can be reduced to a canonical subspace clustering problem. This connection yields provable guarantees: the sample complexity scales with the intrinsic dimension of the data, rather than the ambient dimension, thereby breaking the curse of dimensionality for generalization. I will then turn to the role of representation learning in generalization, using two-layer denoising autoencoders as a tractable model to show that the optimal representations and weight structures differ fundamentally between the memorization and generalization regimes. These results offer a unified perspective on how generative models both learn meaningful structure in latent spaces and synthesize new data in high dimensions. We translate these theoretical insights into practical guidelines for controlled generation, ensuring model safety and privacy. Finally, we conclude by contrasting the generalization performance of diffusion and autoregressive models in the context of state prediction for stochastic dynamical systems. These findings inform new data assimilation methods and provide critical insights across many scientific applications, and establish a foundation for next-generation generative modeling.

Speaker Bio: Qing Qu is an Assistant Professor in EECS at the University of Michigan. He works at the intersection of the foundations of machine learning, numerical optimization, and signal/image processing, with a current focus on the theory of deep generative models and representation learning. Prior to joining Michigan in 2021, he was a Moore–Sloan Data Science Fellow at the Center for Data Science, New York University (2018–2020). He received his Ph.D. in Electrical Engineering from Columbia University in October 2018 and his B.Eng. in Electrical and Computer Engineering from Tsinghua University in July 2011. His work has been recognized with multiple honors, including the Best Student Paper Award at SPARS 2015, a Microsoft PhD Fellowship in Machine Learning (2016), the Best Paper Award at the NeurIPS Diffusion Models Workshop (2023), NSF CAREER Award (2022), Amazon Research Award (AWS AI, 2023), UM CHS Junior Faculty Award (2025), Google Research Scholar Award (2025), and the 1938E Award in Michigan Engineering (2026). He has led and delivered multiple tutorials at ICASSP, CPAL, CVPR, ICCV, and ICML. He was one of the founding organizers and Program Chair for the new Conference on Parsimony & Learning (CPAL), regularly serves as an Area Chair for NeurIPS, ICML, and ICLR, senior area chair for ICASSP’26, and is an Action Editor for TMLR.

We will give an overview of the subject of minimal spectral equipartitions in domains. The first part of the talk will give some history and known results about the related topic of nodal sets of eigenfunctions. The last part of the talk will introduce some recent works with Greg Berkolaiko, Yaiza Canzani, Graham Cox and Peter Kuchment that expand into the world of non-bipartite partitions.  Given time, we’ll discuss implications for graphs in addition to domains.

Abstract: We consider stochastic differential equations that are modified by reward functions or likelihood based weights in order to promote specific events. This perspective applies both to diffusion type models used in generative modeling and to SDEs describing physical phenomena such as molecular dynamics or weather. The main emphasis is on rewards that are non smooth or singular, as they appear in conditioning, threshold objectives, and rare event simulation. We discuss diffusion bridges as a central example, where one seeks typical trajectories connecting prescribed endpoints, for instance during a molecular transition between stable states or between two atmospheric configurations. We also discuss fine tuning of diffusion models with non differentiable rewards, motivated by applications that prioritize tail events and other low probability regions.

Bio: Jakiw Pidstrigach is an AI Research Scientist at Gridmatic. He earned his PhD from the University of Potsdam, with research on filtering and diffusion models. He subsequently held a postdoctoral position at the University of Oxford, where he worked on theory and optimal control of AI.

We discuss aspects of generalized matrix inverses from a "consistency-aware" perspective. We show that many standard tools in engineering and applied mathematics (e.g., the SVD) are commonly mis-applied in ways that undermine solution integrity. We then describe straightforward generalizations of these tools that remedy this situation.

Guidance mechanisms enable controllable generation from diffusion models at inference time. Classifier guidance steers sampling using gradients from a noise-aware classifier, offering principled control but requiring a separately trained network. Classifier-free guidance eliminates the external classifier by interpolating conditional and unconditional predictions, yet demands paired training. Training-free methods such as universal guidance repurpose off-the-shelf networks, but rely on per-step gradient optimization that is expensive and often unstable.

In this talk, I present a general recipe for efficiently steering unconditional diffusion models without gradient guidance during inference. Our approach rests on two structural observations. First, noise alignment: even at early, highly corrupted stages of the reverse process, coarse semantic steering is possible using a lightweight, offline-computed guidance signal—no per-step or per-sample gradients required. Second, transferable concept vectors: a concept direction in activation space, once learned, transfers across both timesteps and samples. A single fixed steering vector learned near low noise levels remains effective when injected at intermediate noise levels for every generation trajectory, providing refined conditional control at negligible cost. These directions are identified via Recursive Feature Machines (RFM), a backpropagation-free feature learning method. Experiments on CIFAR-10, ImageNet, and CelebA demonstrate improved accuracy and generation quality over gradient-based guidance, with significant inference speedups.

The quantum many-body problem lies at the heart of modern physics and chemistry, yet its complexity continues to challenge both theory and computation. In this talk, I will provide a brief introduction to the quantum many-body problem and outline several mathematical questions that may help advance the field. Particular emphasis will be placed on coupled cluster–based approaches, embedding methods, and emerging quantum computational strategies. Throughout the presentation, I will highlight how mathematical analysis and algorithmic development can contribute to improving accuracy, scalability, and conceptual understanding in computational many-body theory.