Math Colloquium - Spring 2026

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall I, unless noted otherwise.

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January 23

Qi Lei, New York University (NYU)

Host: Yan Sun

Virtues and Pitfalls of Weak-to-Strong Generalization: From Intrinsic Dimensions to Spurious Correlations

Weak-to-strong (W2S) generalization is an intriguing paradigm where a strong, pre-trained student model adapts to a downstream task using pseudo-labels generated by a weaker teacher. Despite the apparent limitations of the weak teacher, W2S fine-tuning often leads the student to outperform the teacher itself. This talk will present two recent theoretical perspectives on why and when W2S succeeds.

First, I will discuss the phenomenon through the lens of low intrinsic dimension and in a variance-dominant regime where fine-tuning often takes place in sufficiently expressive low-dimensional subspaces. This analysis reveals a surprising virtue of discrepancy between strong and weak models' feature representation: while variance is inherited in overlapping subspaces, it is dramatically reduced in subspaces of discrepancy, with explicitly derived characterizations of sample complexity and scaling behavior.

Second, I will examine W2S under spurious correlations, a common challenge when labeled data shaping the teacher and unlabeled data guiding the student differ in group proportions. High-dimensional asymptotic analysis reveals that alignment between group distributions is critical: under group-balanced teachers, minority enrichment improves W2S, while under imbalanced teachers, it harms performance. To address this, a simple confidence-based retraining scheme with generalized cross-entropy can mitigate the pitfalls and consistently strengthen W2S across synthetic and real-world datasets.

Together, these works explain why W2S emerges—via intrinsic dimension and representation discrepancy—and how it is affected by spurious correlations, providing a sharper theoretical foundation and guidance for its future development.

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January 30

Kathrin Smetana, Stevens Institute of Technology

Host: Christina Frederick

Randomized model order reduction methods for problems with high-dimensional parameter sets

In this talk, we present randomized algorithms to construct approximations for a set of solutions of parameter-dependent partial differential equations (PDEs), where the parameter set is high dimensional. It is well-known that the Proper Orthogonal Decomposition (POD)/Principal Component Analysis (PCA) and the greedy algorithm perform (quasi-)optimally in approximating such a set. However, both the POD and the greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We suggest breaking the curse of dimensionality by exploiting the concentration of measure phenomenon, which is also sometimes called the "blessing of dimensionality”.

In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.

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February 6

Esteban Tabak, New York University (NYU)

Host: David Shirokoff

The Optimal Transport Barycenter problem as a toolbox for the analysis of data

This talk will discuss a framework for data analysis based on the [Monge] optimal transport barycenter problem (OTBP), which removes from a set of variables any variability that a set of covariates can explain. Solving this problem enables a number of tasks, including conditional density estimation and simulation, factor discovery and transfer learning. We will discuss this framework and its applications, and describe an effective methodology for solving the data-driven OTBP through an adversarial characterization of independence.

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February 13

Harbir Antil, George Mason University

Host: Christina Frederick

Digital Twins, Generative AI, and Beyond: A PDE-Constrained Optimization Perspective

Digital Twins (DTs) are adaptive, real-time virtual replicas of physical systems that integrate physics-based models, sensor data, and intelligent decision-making. At their core, DTs can be rigorously framed within PDE–constrained optimization (PDECO). This talk develops a unified PDECO framework for state estimation and control, leveraging adjoint-based methods in both deterministic and stochastic settings.

To address the challenges of infinite-dimensional, large-scale optimization, we introduce novel function-space trust-region and augmented Lagrangian algorithms, and explore the role of randomized methods in dynamic PDECO.

A central theme is a new connection between PDECO and Generative AI: score-based generative models can be interpreted as backward-in-time PDEs, linking ill-posed inverse problems, stability analysis, and modern machine learning. This perspective bridges physics-informed modeling with data-driven synthesis, opening the door to, for instance, score-based Digital Twins.

Applications span a wide range of domains, including structural and biomedical systems—from bridges and dams to aneurysm modeling, optimal insulation, electromagnetic cloaking, light bending, fusion, and neuromorphic computing. Together, these examples highlight a pathway toward predictive, adaptive, and trustworthy Digital Twins and AI technologies.

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February 20

Guillaume Bal, University of Chicago

Host: Daniel Massatt

Topological insulators and robust edge transport

Surprising asymmetric transport phenomena along interfaces separating insulating bulks have been observed in many areas including electronics, photonics, and geophysics. Such transport, displaying strong robustness to perturbations as an obstruction to Anderson localization, affords a topological origin: systems in the same topological class display similar robust, quantized, interface transport.

This talk considers elliptic partial differential systems. We introduce a topological classification with an explicit computation of the topological invariant, technically the index of a Fredholm operator. We next define a physical observable that quantifies the asymmetric edge transport, but whose practical evaluation is challenging. We then prove a bulk-edge correspondence, a pillar of topological phases of matter, stating that the interface current observable equals the aforementioned topological invariant. We also demonstrate the limitation of the bulk-edge correspondence for non-elliptic systems. The theoretical findings are illustrated with examples ranging from electronics applications to geophysical fluid flows.

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February 27

Clarence Rowley, Princeton University

Host: Alberto Padovan

Learning dynamics: agnostic control and oblique projections

How can one control a system without any knowledge of its dynamics? What might "optimal" control mean in such a setting? How should we balance the competing tasks of learning the dynamics (exploration), and exploiting what we have learned so far? The first part of this talk addresses a recent approach to "agnostic control," where we seek an optimal strategy, based on minimizing the "regret," the difference (or ratio) between the cost of our strategy and the cost incurred by an opponent who knows the dynamics perfectly and plays optimally.

The second part of this talk involves constructing reduced-order models for systems for which an accurate model is known, but is prohibitively complicated (e.g., high-dimensional). In particular, we use ideas from balanced truncation and active subspaces to construct "oblique" (non-orthogonal) projections that incorporate effects of sensitivity in addition to state covariance. We demonstrate and compare these techniques with other methods on a challenging toy problem, and on a nonlinear axisymmetric jet flow simulation with 100,000 state variables.

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March 6

Laura Kubatko, Ohio State University

Host: Kristina Wicke

Composite likelihood approaches to evolutionary inference from genome-scale data

Realistic models for the evolution of genome-scale data involve substantial computational challenges from the perspective of statistical inference. Notably, the likelihood cannot be computed under such models, making inference in the typical statistical frameworks (i.e., the likelihood and Bayesian frameworks) intractable. In this talk, the composite likelihood will be used as a tool to enable statistical inference in a firm theoretical framework. In particular, I will show that the composite likelihood can be used to obtain statistically-consistent estimates of speciation times on a fixed phylogenetic tree and can be used in a Bayesian Markov chain Monte Carlo (MCMC) algorithm to obtain interval estimates of the speciation times that achieve the correct coverage. Finally, I show how a simulated annealing approach can be used to search for the optimal phylogeny for a given data set. These applications highlight the usefulness of the composite likelihood approach in cases where the likelihood is computationally intractable.

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March 13

Bard Ermentrout, University of Pittsburgh

Host: James MacLaurin

Riding the neural waves: Mathematics of trigger & phase waves in neural media

Recent improvement in technology have enabled neuroscientists to simultaneously record activity of many neurons at high spatial and temporal resolution. This has allowed them to discover that activity is organized into a variety of spatial patterns such as plane waves, bullseyes, and rotating waves. In this talk, I want to distinguish two different classes of wave-like activity: (1) evoked waves or "trigger waves", and (2) phase waves. In the former, the onset of activity in one area requires prior activity in a neighboring area, while in the latter, the apparent wave motion is a consequence of timing differences between area. I will present some recent results on the role of inhibition in controlling the propagation and stability of trigger waves. Next, I will consider coupled phase equations that describe spatio-temporal activity in intrinsically oscillatory media. I will describe recent work where we are able to extract hidden waves from human cortical recordings. Finally, I will present some work showing how ongoing phase waves can promote the propagation of trigger waves in an anisotropic manner.

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