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.

Pure ring maps arise naturally in algebraic geometry and commutative algebra. For example, inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps are all pure. It is useful to know when a map is pure because pure maps satisfy effective descent properties and many classes of singularities are preserved under the operation of taking pure subrings. For example, a theorem of Boutot (for rings of finite type over a field) and myself (in general) says that for Noetherian \(\mathbf Q\)-algebras, pure subrings of rings with (at worst) rational singularities have (at worst) rational singularities. Such results are often called Boutot-type theorems.

In this talk, I will discuss a new class of pure ring maps which arise geometrically from families of varieties of the same dimension. In fact, this yields a characterization of the splinter property: A Noetherian ring \(R\) is a splinter (i.e., all module-finite extensions \(R \to S\) split) if and only if every locally equidimensional surjective morphism \(Spec(S) \to Spec(R)\) is pure. Since not all pure maps are locally equidimensional and Boutot-type theorems fail for some classes of singularities like \(F\)-rationality, this raises the question: Are there "weak" Boutot-type theorems for pure ring maps that are also equidimensional? I will discuss my affirmative solution to this question for \(F\)-rationality.

We revisit the problem of estimating the fundamental matrix of a pair of perspective cameras, a cornerstone of geometric computer vision. As is well-known, linear solvers require at least 8 point correspondences, whereas nonlinear minimal solvers require just 7 in the uncalibrated case or 5 in the calibrated case. In this paper, we consider a special case of the 7-point problem where 5 of the points are configured to lie on two lines, which has previously been shown to have a unique solution. As a theoretical contribution, we offer an analysis of how this uniqueness manifests in the standard 7-point algorithm. On a practical level, we provide the first practical linear solver for the minimal problem associated to this special configuration. Additionally, we evaluate a heuristic 5-point fundamental matrix solver based on the construction of virtual midpoints. When combined with early non-minimal fitting, the runtime and accuracy of our solver is competitive with the state-of-the-art on multiple benchmarks. 

In this talk we will discuss old and not so old inequalities on the
volume of the orthogonal projections (sometimes called local
Loomis-Whitney type estimates).  We will explore connections of those
inequalities to inequalities for mixed volumes as well as inequalities
of the Minkwoski sums of convex bodies.
 

Given a finitely generated module \(M\) over a noetherian local ring \(R\), one may assign to it a conical affine variety, called the cohomological support variety of \(M\) over \(R\). This theory was first developed by Luchezar Avramov for local complete intersection rings in 1989, and by the work of many has recently been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode important homological information about \(M\) as well as \(R\). In this talk I will discuss what cohomological support varieties are, why they are useful, and some recent work on how they behave when restricting along a local homomorphism.