Math Colloquium - Spring 2026

To join the Applied Mathematics Colloquium mailing list visit https://groups.google.com/a/njit.edu/forum/?hl=en#!forum/math-colloquium/join (Google Profile required). To join the mailing list without a Google Profile, submit the seminar request form.

*******************************************************************************************************

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.

*******************************************************************************************************

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.

*******************************************************************************************************

February 6

Esteban Tabak, New York University (NYU)

Host: David Shirokoff

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

*******************************************************************************************************

February 13

Harbir Antil, George Mason University

Host: Christina Frederick

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

*******************************************************************************************************

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.

*******************************************************************************************************

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.

*******************************************************************************************************

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.

*******************************************************************************************************

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.

*******************************************************************************************************

March 27

Nestor Guilen, Texas State University

Host: Christina Frederick

The many roles of Fisher information in kinetic theory

The Fisher information is a fundamental object in statistical inference that has been applied fruitfully to the analysis of dissipative equations like the Fokker-Planck equation / Langevin dynamics, both to understand convergence to equilibrium and understand the formation (or not!) of singularities. I will first discuss some of these classical uses of the Fisher information as well as some history of how this tool made its way to PDE theory, and then I will present recent work with Luis Silvestre where we found the Fisher information is monotone in time (and thus, provides an arrow of time) for the Landau equation from plasma physics. This result made it possible to resolve the long standing problem of blow up for the space homogeneous Landau equation, which was subsequently extended to the homogeneous Boltzmann equation by Imbert, Silvestre, and Villani.

*******************************************************************************************************

April 10

Ben Raphael, Princeton University

Host: Kristina Wicke

Spatiotemporal dynamics of biological tissues

The spatial organization of tissues is essential to their biological function. Recent spatial technologies yield high-throughput and high-dimensional measurements of RNA, proteins, metabolites, and other modalities at thousands of locations within tissue sections, revealing spatial patterns of cell types and molecular activity. However, current datasets are often sparse and incomplete due to technological and cost constraints. I will present mathematical and machine learning approaches to overcome this sparsity by modeling the latent geometry of individual tissue slices and by integrating measurements across multiple modalities over space and time. These approaches utilize implicit neural representations, optimal transport, and related techniques. We apply the resulting methods to analyze spatial variation in cell types and gene expression in normal tissues, derive gene expression gradients within tumor microenvironments, reconstruct three-dimensional tissue architecture across modalities, and describe spatiotemporal changes in expression during organismal development.

*******************************************************************************************************

April 17

Zecheng Zhang, University of Notre Dame

Host: Xinyu Zhao

Operator learning approximation and its distributed theoretical extension

In recent years, Operator Learning (OL) has emerged as a transformative tool in large-scale scientific computing, demonstrating remarkable success in complex applications such as climate modeling. Despite these empirical triumphs, the underlying mathematical framework remains in its infancy.

In this talk, we present a unified neural operator approximation framework capable of formulating several popular OL architectures. We provide a rigorous analysis of error convergence rates, specifically examining the interplay between discretization size, network depth/width, and the number of basis functions. Furthermore, we extend this theory to distributed settings, addressing the challenges of multiscale and multi-operator approximation. We conclude by discussing the future of the field: multi-operator learning frameworks that approximate multiple operators using the same network. This idea aims to handle the extrapolation issue of the ML and we will focus on a theoretical study of one framework.

*******************************************************************************************************

April 24

Jeremy Marzuola, University of North Carolina Chapel Hill

Host: Amir Sagiv

Recent advances in equipartitions of domains

We will give an overview of the subject of minimal spectral equipartitions in do- mains. 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.

*******************************************************************************************************

May 1

Hernan Makse, City College of New York (CCNY)

Host: Lou Kondic

Title/Abstract Forthcoming

*******************************************************************************************************

Last Updated: April 8, 2026