When a diffusion model is not memorizing the training samples, what does it generate, and why? In this talk, I will describe a quantitative framework for understanding the distribution produced by a learned diffusion model through a data-driven geometric object: a log-density ridge manifold of the smoothed training distribution. This manifold acts as a backbone for generation and reveals a three-stage inference behavior: trajectories first reach the ridge, then align in normal directions, and finally slide along tangent directions. 

This perspective allows us to quantify how training error influences generation in different directions, and to explain when inter-mode generations arise. I will also present a random feature example in which the model’s inductive bias can be decomposed explicitly into architectural bias and optimization error, and tracked along the inference dynamics. Experiments on synthetic multimodal distributions and MNIST latent diffusion support the theory in both low- and high-dimensional settings.

The goal of this talk is to consider two instances of a class of reconstruction problems that aim to recover an unknown signal from indirect measurements that are algebraic in nature. Such problems are paramount in mathematics, enjoying applications in a wide array of fields like molecular imaging, machine learning, and geo-positioning. In this talk, we will motivate and study the generic crystallographic phase retrieval problem and the orthogonal beltway problem and deduce conditions in each setting that guarantee signal recovery. In the latter problem, we will also develop a polynomial-time algorithm that recovers the signal while remaining robust to small amounts of noise. We resolve both of the central problems of this talk by recasting them as special instances of the problem of recovering a signal from its second moment under the multi-reference alignment (MRA) model, for which a rich theory has been developed in prior literature.

Abstract: Given a simplicial projective toric variety with at most one singular point, we study the asymptotic structure of the sumsets arising from its parametrization. We introduce a notion of sumsets regularity and relate it to the Castelnuovo–Mumford regularity of the variety, translating an algebraic problem into a combinatorial one. We then obtain new upper bounds on the sumset regularity, and hence on the Castelnuovo–Mumford regularity of these varieties. This talk is based on joint work with Ignacio García-Marco and Philippe Gimenez. 

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.