Math Colloquium - Fall 2025

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September 5

Alberto Padovan, NJIT (Mechanical and Industrial Engineering)

Host: Michael Siegel

Leveraging oblique projections for model reduction: recent advances in Petrov-Galerkin and non-intrusive formulations

Developing accurate reduced-order models (ROMs) for high-dimensional physical systems (e.g., fluids, solids, systems in thermochemical kinetics, etc.) is essential to accelerate simulations and enable engineering tasks like design and optimization. In this talk, we emphasize the importance of oblique projections for the development of ROMs with long-time predictive accuracy, and we present recent advances on their computation using both intrusive
and non-intrusive methods. 

Several systems in engineering, including transport-dominated fluid flows, exhibit high sensitivity to low-energy coordinates. These coordinates are typically truncated in ROMs obtained with orthogonal projections designed to capture the high-energy features of the system. As a consequence of this truncation, the resulting ROMs can fail to faithfully reproduce the dynamics of the original system. By contrast, oblique projections can be used to compute ROMs that capture the energetic features of the flow while also accounting for the low-energy coordinates that are necessary to accurately describe the dynamics of the full-order model. In this presentation, we present two novel formulations to compute oblique-projection-based (or Petrov-Galerkin) models for nonlinear systems. The first one is similar in spirit to the well-known balanced truncation formulation for linear systems, and it identifies a quasi-optimal oblique projector by balancing the state and gradient covariance matrices associated with the flow of the underlying full-order model. The second method is a refinement of the latter, where oblique projections for Petrov-Galerkin model reduction are further optimized against high-fidelity data to reduce forecasting error. While these two methods can work very well and outperform existing model reduction formulation, they are highly intrusive and require direct access to the underlying governing equations. They may therefore be inapplicable to systems that are simulated using black-box solvers (e.g., commercial software or legacy codes) whose source code might be proprietary or very difficult to modify. To address this issue, we introduce a third (non-intrusive)  formulation, which we call “NiTROM: Non-intrusive Trajectory-based optimization of ROMs.” Specifically, given high-fidelity data from a black-box solver, NiTROM computes a ROM by simultaneously seeking the optimal projection and latent-space dynamics that minimize forecasting error. This formulation can achieve predictive accuracy comparable to that of the first two (intrusive) Petrov-Galekin methods, and it has been shown that it can significanly outperform state-of-the-art non-intrusive frameworks like Operator Inference. All the methods discussed in this talk are demonstrated on several systems, including incompressible fluid flows with states.

September 12

Anand Oza, NJIT

Waves and Interaction Modes of Capillary Surfers

We present the results of a combined experimental and theoretical investigation into “capillary surfers,” which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. Recent experiments showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers’ horizontal positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good quantitative agreement with experimental data. We then study the vertical dynamics of a floating disc, recasting the elliptic boundary value problem with mixed boundary conditions as an integral equation that is solved numerically. We find that the disc exhibits resonant oscillations at a critical value of the forcing frequency, in agreement with experiment. Generally, our work shows that self-propelling objects coupled by interfacial flows constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant.

September 19

Bruce Pitman, University of Buffalo

Host: Yuan-Nan Young

Inverse Problems via Gaussian Processes

One class of inverse problems consists of determining the properties of an inhomogeneous inclusion in an otherwise homogeneous domain from measurements at the domain boundary. This is the setting of imaging problems in geophysics and biomedicine. This inverse problem can be complicated to solve, and specialized methods have been developed to that end. Here we report on a different approach to solving the inverse problem, using Gaussian Process emulation. This emulation provides a fast solver with very modest computational cost. We illustrate the idea with three simple applications.

SPECIAL SEMINAR - September 25

Eliot Fried, Mechanics and Materials Unit Okinawa Institute of Science and Technology 

Host: Jointly by Mechanical Engineering and Mathematical Sciences. For details, please contact Anthony Rosato, anthony.rosato@njit.edu

Location: Tiernan Hall, Lecture Hall 1 (10:00AM - 11:00AM)

Inside out without effort: Exceptional linkages and the geometry of isometric band eversions

We explore a unified framework linking the kinematics of underconstrained ring linkages to the isometric and isoenergetic eversion of continuous bands. The starting point is the M¨obius kaleidocycle—a closed chain of n ≥ 7 identical rigid links joined by revolute hinges at a critical twist angle. Despite being spatially underconstrained, such linkages exhibit only a single internal degree of freedom, manifested in a smooth everting motion. As n → ∞, this family of linkages converges to a ruled M¨obius band with three half-twists, uniform torsion midline, and trefoil-knotted boundary. Motivated by this limit surface, we consider the inverse problem: determining all stable isometric deformations from circular helicoids to M¨obius bands, and characterizing their shape-preserving periodic eversions. For isotropic elastic bands with bending energy depending only on mean curvature, these motions are isoenergetic, requiring no work input. Finally, we return to discrete systems by showing how the midline of such a band can be discretized via a nonuniform finite-difference scheme, yielding a class of linkages that preserve mobility for arbitrary link arrangements. This synthesis reveals deep geometric and mechanical connections between exceptional-mobility linkages and continuous surfaces capable of effortless eversion, with implications for deployable structures, robotics, and the design of topologically nontrivial elastic systems.

Biographical Sketch: The speaker received his Ph.D. in Applied Mechanics from the California Institute of Technology in 1991 and held a National Science Foundation postdoctoral fellowship. He has since held tenured faculty positions at the University of Illinois at Urbana–Champaign, Washington University in St. Louis, McGill University—where he was Tier I Canada Research Chair in Interfacial and Defect Mechanics—and the University of Washington. He is now Professor and Head of the Mechanics and Materials Unit at the Okinawa Institute of Science and Technology. His research spans mechanics, thermodynamics, and geometry, with current interests in surface instabilities, reaction–diffusion processes coupled to bulk strain, and underconstrained linkages. He is also co-inventor on two U.S. patents

September 26

Jingyi Jessica Li, Fred Hutchinson Cancer Center 

Host: Chenlu Shi

Nullstrap: A Simple, High-Power, and Fast Framework for FDR Control in Variable Selection for Diverse High-Dimensional Model

Balancing false discovery rate (FDR) control with high statistical power remains a central challenge in high-dimensional variable selection. While several FDR-controlling methods have been proposed, many degrade the original data—by adding knockoff variables or splitting the data—which often leads to substantial power loss and hampers detection of true signals. We introduce Nullstrap, a novel framework that controls FDR without altering the original data. Nullstrap generates synthetic null data by fitting a null model under the global null hypothesis that no variables are important. It then applies the same estimation procedure in parallel to both the original and synthetic data. This parallel approach mirrors that of the classical likelihood ratio test, making Nullstrap its numerical analog. By adjusting the synthetic null coefficient estimates through a data-driven correction procedure, Nullstrap identifies important variables while controlling the FDR. We provide theoretical guarantees for asymptotic FDR control at any desired level and show that power converges to one in probability. Nullstrap is simple to implement and broadly applicable to high-dimensional linear models, generalized linear models, Cox models, and Gaussian graphical models. Simulations and real-data applications show that Nullstrap achieves robust FDR control and consistently outperforms leading methods in both power and efficiency.

October 3

Matt Holzer, George Mason University 

Host: James MacLaurin

Persistence of steady-states for dynamical systems on large networks

The goal of this work is to identify steady-state solutions to dynamical systems defined on large, random families of networks. We do so by passing to a continuum limit where the adjacency matrix is replaced by a non-local operator with kernel called a graphon. Our work establishes a rigorous connection between steady-states of the continuum and network systems. Precisely, we show that if the graphon equation has a steady-state solution whose linearization is invertible, there exists related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon. The proof involves setting up a Newton--Kantorovich type iteration scheme which is shown to be a contraction on a suitable metric space. We also discuss linear stability properties that carry over from the graphon system to the graph dynamical system. Our results are applied to twisted states in a Kuramoto model of coupled oscillators, steady-states in a model of neuronal network activity, and a Lotka--Volterra model of ecological interaction. We also discuss subsequent extensions of our work to nonhyperbolic points with saddle-node bifurcations leading to the onset of synchrony as our primary motivating example.  

October 10

Liliana Borcea, Columbia University 

Host: Amir Sagiv

Data driven reduced order modeling for first order hyperbolic systems with application to waveform inversion

Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, variable coefficients. The traditional formulation of the inverse problem, called full waveform inversion (FWI), estimates the unknown coefficients via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to "learn" a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves. 

October 17

Ricardo Baptista, University of Toronto

Host: Amir Sagiv

Memorization and Regularization in Generative Diffusion Models

Diffusion models have emerged as a powerful framework for generative modeling in the information sciences and many scientific domains. To generate samples from the target distribution, these models rely on learning the gradient of the data distribution's log-density using a score matching procedure. A key element for the success of diffusion models is that the optimal score function is not identified when solving the denoising score matching problem. In fact, the optimal score in both unconditioned and conditioned settings leads to a diffusion model that returns to the training samples and effectively memorizes the data distribution. In this presentation, we study the dynamical system associated with the optimal score and describe its long-term behavior relative to the training samples. Lastly, we show the effect of two forms of score function regularization on avoiding memorization: restricting the score's approximation space and early stopping of the training process. These results are numerically validated using distributions with and without densities including image-based inverse problems for scientific machine learning applications.

October 24

Marian Gidea, Yeshiva University

Host: Jonathan Jaquette

Modeling the dynamics of piezoelectric energy harvesters

Ambient energy harvesting by autonomous electro–mechanical systems is a potential source of renewable energy. We consider some models of  energy harvesting devices consisting of cantilever  beams under ambient vibrations. Piezoelectric layers are attached to the beams, serving to convert mechanical energy into electrical energy. We show some results of numerical simulations as well as of lab experiments. We also present a model based on a Hamiltonian system subject to dissipation and periodic forcing, and describe a mechanism that leads to energy growth over time.

October 31

Shari Moskow, Drexel University

Host: Thi-Phong Nguyen

On optimality and bounds for internal solutions generated from impedance data driven Gramians

We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming the background snapshots for a known background coefficient using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data only for a time domain plasma wave equation with an unknown potential q. We use this to show convergence for general unknown $q$ in one dimension. We show numerical experiments and applications to SAR imaging in higher dimensions.  

Druskin, V., Moskow, S., Zaslavsky, M. 

November 7

Edo Kussell, New York University

Host: James MacLaurin

Evolutionary Phase Transitions in Fluctuating Environments

Bacteria possess a very large toolbox of molecular mechanisms, coded by specific genes, which enable survival under stressful conditions. These mechanisms on their own are often inefficient or costly, and only by using them strategically do bacteria gain a long-term advantage. This talk examines the strategies that bacteria use to regulate their survival toolbox. By encoding and using memory in different ways, bacteria can optimize their long-term growth potential. This optimization can be understood by a statistical mechanics analogy. I describe a phase diagram structure in which memory levels are optimized as a function of the statistics of a randomly fluctuating environment, and bacterial survival strategies can undergo different types of phase transitions. We are currently studying this phase diagram experimentally, and I will discuss some of our recent experiments quantifying costs and benefits of gene regulation in fluctuating environments.

November 14

Susan Minkoff, Brookhaven National Laboratory

Host: Amir Sagiv

Digital Twins for Wind Energy and Leading Edge Erosion Detection

One of the main sources of renewable energy is wind, which generates tremendous power while also reducing the need for greenhouse gas-emitting traditional power sources such as hydrocarbons and coal. However, many wind turbines installed in the early 2000's are nearing the end of their lifespan, and the problem remains of how to maintain, reduce, or decommission these aging turbines in a cost efficient way. In this talk we describe a digital twin for damage detection and maintenance scheduling of wind turbines which can track the condition of a wind turbine under different operating conditions. A key concern for wind energy that contributes to power production losses and high maintenance costs is deterioration of the turbine blades over time from environmental stressors such as lighting strikes, icing, and accumulation of airborne particles which can result in leading edge erosion of the blades. We will discuss surrogate modeling of the turbines and classification of levels of leading edge erosion via machine learning.

Sue Minkoff (Brookhaven National Lab), Aidan Gettemy, Todd Griffith, Ipsita Mishra (University of Texas at Dallas), John Zweck (New York Institute of Technology), and Elaine Spiller (Marquette University)

November 21

Marco de Paoli, TU Wien

Host: Lou Kondic

Flow morphology in porous media convection: how heat and mass transport shapes flow structures

Convective mixing in porous media plays a central role in many natural and engineered systems - from underground carbon dioxide sequestration to the migration of contaminants in groundwater. The coupling between convection and diffusion produces intricate, time-evolving flow patterns that govern how heat and chemical species are transported and mixed.

In this seminar, we will explore how these complex flow structures emerge and evolve. The first part will focus on modeling heat and mass transport in porous-media convection and developing physical models relevant to geophysical processes. In the second part, we will introduce tools from topological data analysis, in particular persistent homology, to study classical flow configurations such as one-sided and Rayleigh-Bénard convection. These methods allow us to quantify flow morphology and reveal new relationships between macroscopic transport properties and the underlying topology of the flow.

Drawing on a large dataset of numerical simulations and previous studies, we will discuss how variations in heat transport correlate with changes in flow structure across a broad range of governing parameters. The talk will highlight how combining mathematical modeling, numerical simulation, and topology can lead to deeper insights into convective processes in porous systems.

December 5

Vera Mikyoung Hur, University of Illinois Urbana-Champaign

Host: Xinyu Zhao

Title/Abstract Forthcoming

Last Updated: October 22, 2025