Schedule

Abstract: Many approaches to machine learning have struggled with applications that possess complex process dynamics. I will describe an approach to machine learning of dynamical systems based on Koopman Operator Theory (KOT) that produces generative, predictive, context-aware models amenable to (feedback) control applications. KOT has deep mathematical roots and I will discuss its basic tenets. Its first applications were in fluid mechanics and a number of these will be showcased a number of these, but a number of other examples will be discussed, including use in soft robotics.

Acknowledgement: Support from ARO, AFOSR, DARPA, NSF and ONR is gratefully acknowledged.

Abstract: We study the problem of inferring interaction kernels from observed behaviors in particle and agent-based systems, which arise in fields ranging from physics to the social sciences. We first consider stochastic systems whose interaction kernels depend on pairwise distances, and introduce a nonparametric inference framework based on a regularized maximum likelihood estimator. This approach enables the estimation of distance-based interaction kernels with consistency and a near-optimal convergence rate that is independent of the dimension of the state space. We also analyze the error induced by discrete-time observations and demonstrate the effectiveness of the method through numerical experiments on models such as stochastic opinion dynamics and the Lennard–Jones system. Finally, we extend the analysis to the identification of nonlocal interaction potentials in aggregation–diffusion equations from noisy data using sparsity-promoting methods. This talk is based on joint work with Fei Lu, Mauro Maggioni, José A. Carrillo, Gissell Estrada-Rodriguez, and László Miklós.

10:50 AM - 11:10 AM: Mykhailo Potomkin 

Multiscale Analysis of Accumulation Dynamics in Living Systems

Abstract: Multi-agent living systems exhibit unique dynamics such as accumulation at confinements, where their behavior cannot be reduced to simple sedimentation, but instead involves tangential transport along boundaries and escape mechanisms. A population-level description of such systems is therefore challenging: confinement induces a boundary layer, while accumulation dynamics requires an additional evolution partial differential equation on the confining surface, rather than conventional boundary conditions. In this talk, I will present a kinetic framework that enables the direct computation of probability distribution functions for such active systems. A distinguishing aspect of this approach is its explicit treatment of wall accumulation through the use of two coupled probability distribution functions: one describing the bulk population and the other representing rods accumulated at the boundary. Another novel feature is the structure of the governing equation, which is degenerate: it is second-order in one spatial variable and first-order in another. The main focus of this talk is the rigorous justification of this model via multiscale analysis.

11:10 AM - 11:30 AM: Sayun Mao

Optimal Reward Scheduling for Contingency Management: A Probabilistic Model of Drug Relapse with Temporal Discounting

Abstract: Contingency management (CM) is an effective behavioral intervention for substance use disorders in which patients receive rewards for negative drug tests, but its high cost limits wider adoption. We propose a mathematical model in which the relapse rate of substance use is suppressed by a sequence of rewards conditioned on abstinence. Expectation of larger future rewards reduces relapse rates by compensating for drug cravings. However, the discounting of anticipated rewards mitigates the risk suppression. Reward schedule design can be formulated as an optimization problem that minimizes relapse probability under a fixed total budget. Our analysis shows that incrementally increasing voucher values can outperform constant rewards when aligned with individuals' discounting behavior, and that adapting rewards to changes in craving further improves outcomes. We further explore how these strategies perform across diverse populations, providing guidance in the design of more cost-effective CM protocols.

11:30 AM - 11:50 AM: Mohammad Yasir Feroz Khan

Maximizing Reaction via Spatial Localization

Abstract: We study a PDE-constrained variational problem modeling spatial distributions of enzymes that maximize their reaction with substrates in biological cells. Given an admissible enzyme concentration, the substrate concentration is defined as the solution of a reaction-diffusion equation with prescribed boundary data.  The objective functional, representing total reaction flux, is a nonlinear functional of the coupled enzyme-substrate system. We construct reaction-maximizing sequences through localization of enzymes, calculate the first and second variations of the reaction functional, and show the nonexistence of local and global maximizers. To address this ill-posedness, we introduce regularized reaction functionals and study their basic variational properties. We conclude with a discussion of the biological interpretation of these results.

11:50 AM - 12:10 PM: Xinyi Lu

Mathematical and Computational Modelling of Breast Cancer Initiation

Abstract: Mutations in the tumor suppressor BRCA1 greatly increase lifetime breast cancer risk. In addition to its role in genomic instability, experimental studies suggest that BRCA1-mutant mammary epithelial cells may become abnormally sensitive to hormonal cues, display pregnancy-like side-branching programs outside pregnancy, and accumulate basal--luminal intermediate (BLI) states during premalignant progression. At the same time, altered Activin/Follistatin signaling and disrupted epithelial organization point to a growth-control problem that spans lineage dynamics, biochemical inhibition, and tissue architecture. However, how these features combine to permit tumor initiation remains unclear. Here, we develop a chemo-mechanical feedback mathematical model in the breast duct to  generate hypotheses of what causes cancer initiation of the presence of BRCA1 mutation.

10:50 AM - 11:10 AM: Adrien Weihs

Scaling Laws for Multiple Operator Learning

Abstract: The multiple operator learning problem is central to PDE foundation models, where the goal is to learn not just a single operator, but a family of related operators indexed by parameters, tasks, or physical regimes. In this talk, I will present a novel framework for this setting, called Multiple Neural Operators, together with empirical results illustrating its effectiveness. I will also discuss recent theoretical advances establishing scaling laws for expressivity and generalization, clarifying how model size, architecture, and sampling govern approximation power and predictive performance. Together, these results provide a foundation for the analysis and design of scalable learning systems for operator families.

11:10 AM - 11:30 AM: Botao Jin

Deep Signature Approach for McKean-Vlasov FBSDEs in a Random Environment

Abstract: Mean-field games with common noise provide a powerful framework for modeling the collective behavior of large populations subject to shared randomness, such as systemic risk in finance or environmental shocks in economics. These problems can be reformulated as McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) in a random environment, where the coefficients depend on the conditional law of the state given the common noise. Existing numerical methods, however, are largely limited to cases where interactions depend only on expectations or low-order moments, and therefore cannot address the general setting of full distributional dependence. In this work, we introduce a deep learning-based algorithm for solving MV-FBSDEs with common noise and general mean-field interactions. Building on fictitious play, our method iteratively solves conditional FBSDEs with fixed distributions, where the conditional law is efficiently represented using signatures, and then updates the distribution through supervised learning. Deep neural networks are employed both to solve the conditional FBSDEs and to approximate the distribution-dependent coefficients, enabling scalability to high-dimensional problems. Under suitable assumptions, we establish convergence in terms of the fictitious play iterations, with error controlled by the supervised learning step. Numerical experiments, including a distribution-dependent mean-field game with common noise, demonstrate the effectiveness of the proposed approach.

11:30 AM - 11:50 AM: Alexander Sietsema

Harmful Overfitting in Sobolev Spaces

Abstract: Motivated by recent work on benign overfitting in overparameterized machine learning, we study the generalization behavior of functions in Sobolev spaces Wk,p(Rd) that perfectly fit a noisy training data set. Under assumptions of label noise and sufficient regularity in the data distribution, we show that approximately norm-minimizing interpolators, which are canonical solutions selected by smoothness bias, exhibit harmful overfitting: even as the training sample size n → ∞, the generalization error remains bounded below by a positive constant with high probability. Our results hold for arbitrary values of p ∈[1,∞), in contrast to prior results studying the Hilbert space case (p = 2) using kernel methods. Our proof uses a geometric argument which identifies harmful neighborhoods of the training data using Sobolev inequalities.

11:50 AM - 12:10 PM: Zeyi Xu

Accelerating Sinkhorn via Dual Mirror Descent

Abstract: Why does Sinkhorn work so well—and can we make it faster? We answer both questions by revealing that Sinkhorn is exactly dual mirror descent with a mirror function tuned to the geometry of optimal transport, achieving dual relative smoothness with constant $L= 1$. This geometric insight directly motivates AccSinkhorn, an accelerated algorithm that inherits Sinkhorn’s per-iteration cost while provably improving the convergence rate from $O(1/k)$ to $O(1/k^2)$. On synthetic and real-world optimal transport problems, Acc-Sinkhorn converges more than 10×faster than Sinkhorn at small regularization, the regime where high precision is required and where Sinkhorn struggles most.

10:50 AM - 11:10 AM: Xu-Hui Zhou

Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks

Abstract: Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the standard EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well-behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics-informed transfer learning, and demonstrate that in so-doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the standard EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with an inviscid Burgers' equation, Sod's shock tube, and a two-dimensional blast wave.

11:10 AM - 11:30 AM: Hesham Morgan

AI-Powered Monitoring of Afforestation Using Multispectral Drones and Deep Learning

Abstract: We present an advanced deep learning framework for evaluating afforestation success in arid environments, focusing on tree classification using multispectral drone imagery. Conducted within the 44,000 Trees Project in Shuayb Al-Budai, Saudi Arabia, our study deployed the MicaSense Altum-PT sensor to capture high-resolution, multispectral and thermal imagery across a 5 km² planted area. After detecting over 43,000 circular planting pits (Level 1), we applied a Level 2 classification model using Faster R-CNN with a ResNet101 backbone to determine vegetation status inside each pit: No Tree, Small Tree, or Large Tree. Trained on 4,500 annotated samples and enhanced with NDVI, pan-sharpened bands, and thermal data, the Level 2 model achieved 91.8% overall accuracy. Spatial analysis revealed that 50.6% of pits contained small trees, 26.2% large trees, and 23.2% had no growth. Challenges such as cloud shadows, soil reflectance heterogeneity, and early-stage vegetation were addressed through spectral preprocessing, zone-wise validation, and augmentation techniques. This classification not only provided quantitative insights into seedling survival and growth variability but also enabled the production of actionable shapefiles for guiding irrigation, replanting, and field validation strategies. Our work demonstrates the power of AI-integrated remote sensing for high-precision ecological monitoring and offers a replicable model for afforestation tracking under extreme climatic constraints, directly supporting SDG 15 and Saudi Vision 2030 goals.

11:30 AM - 11:50 AM: Jesus Quiros

Deep Network Control for Ring-Shaped Circulation for Wheeled Mobile Robots based on Van der Pol Limit-Cycle Reference

Abstract: Ring‑shaped circulation is a form of limit‑cycle motion that remains difficult to stabilize in mobile robotic systems. To address this challenge, this paper introduces a nonlinear trajectory‑generation method combined with a learning‑based control framework that enables a wheeled mobile robot to maintain stable circular motion. A Van der Pol-type oscillator is employed as a dynamic reference model whose asymptotically stable limit cycle defines the desired ring‑shaped path. In contrast to purely geometric circular references, the oscillator provides intrinsic orbit attractivity, reducing large transient deviations and ensuring smooth convergence toward the desired radius. The path-following problem is formulated as a reinforcement learning task. A Deep Q-Network controller learns a discrete velocity policy based on the vehicle’s relative configuration with respect to the nonlinear reference, characterized by the Euclidean distance and trigonometric orientation error. Unlike conventional geometric or Lyapunov-based path-following strategies, the proposed approach directly learns a stabilizing control policy capable of starting from any initial condition without explicit model inversion.  Moreover, we establish boundedness and orbital convergence properties of the closed-loop system relative to the reference limit cycle. Simulation and experimental results validate the effectiveness and robustness of the learned policy in achieving sustained ring-shaped circulation.

10:50 AM - 11:10 AM: Manuchehr Aminian

The Everything Bagel of Applied Math

Abstract: We present ongoing research in the field of sparse discovery of differential equations from data. We investigate the applicability of these tools in the context of compartmental models; for which discovery also requires conservation constraints on the state variables. In the second half of the talk, I will present my efforts, inspired by this field, building and teaching a special topics course toward a capstone course for our applied mathematics undergraduate students relevant for "modern day applied math." As I will show, a little bit of everything is necessary, and early feedback has been positive among the students. I will give some reflection of overall structure and adjustments for future iterations.

11:10 AM - 11:30 AM: Ngoc Kim Ngan Tran

A Comprehensive Study of Large Language Models News Bias Detection versus Crowdsourced Human Labels

Abstract: Large Language Models (LLMs) are increasingly used in news summarization and content moderation, but whether they align with human perceptions of bias remains a critical open question. This study assesses the performance of leading models, including Gemini, GPT, Llama, Qwen3, Gemma3, and R1-1776, on a dataset composed of subjective labels serving as a ‘ground truth’ of bias perception of 290 articles from Fox News (117), CNN (63), and the BBC (110), each labeled by Amazon Mechanical Turk workers. Our experiment conducts a systematic analysis across seven contextual tiers, examining how adding metadata, such as news source, reader political affiliation, and demographic details (age and gender), influences both human and LLM perceptions of bias. We utilize post-inference Chain of Thought (CoT) analysis to understand how different LLMs produce their assessments and evaluate the accuracy of these assessments. We explore whether models share a common definition of bias across architectures or diverge into model-specific perspectives. This presentation will highlight which models most closely match human consensus and identify which particular metadata most strongly influence shifts in AI judgment. We also analyze how well the models’ assessments align with each other. By examining internal thought processes and the justifications for their results, we uncover the mechanics of AI bias detection and assess whether a model’s public explanation truly reflects its internal reasoning.

11:30 AM - 11:50 AM: Mia Zender

DeGroot Dynamics on Multilayer Networks with Directed and Acyclic Interlayer Connectivities

Abstract: Traditional opinion dynamics models typically analyze how an individual's opinion on one topic is affected by the opinions of their neighbors. However, people's opinions on one topic are also influenced by their opinions on related topics. In an election, for example, a voter will consider their opinions on the important issues in the election to form an opinion of each candidate. To consider this additional influence, we introduce a multilayer version of the traditional DeGroot model. In contrast to the other multilayer DeGroot models, our model requires that the edges between layers in our network be directed and acyclic, where an individual's opinion on one topic is influenced by their opinions on multiple other topics, as well as the opinions of their neighbors. In this talk, I will introduce our model and give an overview of our proof that this model converges under the condition that every individual takes their own opinion into account when updating their opinion on each topic. I will also summarize our proof that a specific network structure in some of the layers implies eventual consensus in each layer. Additionally, I will show some numerical simulations of this model on a two-layer network and discuss how consensus and polarity change with different network structures and different types of initial data in each layer.

11:50 AM - 12:10 PM: Eric Adams

Causal Analysis of One-Year Cardiovascular Safety of Combination Versus Stimulant ADHD Therapy Using the MIMIC-IV Database

Abstract: 

BACKGROUND: Cardiovascular safety is an important consideration in the treatment of attention-deficit/hyperactivity disorder (ADHD), especially when comparing stimulant-only therapy with combination therapy involving both stimulant and non-stimulant medications. Although both approaches are used in practice, adjusted evidence comparing their cardiovascular risk remains limited. The analytic cohort included 550 patients, of whom 83 received combination therapy and 467 received stimulant-only therapy. Within 1 year, 26 patients experienced the cardiovascular composite outcome.

METHODS : We estimated the causal effect of baseline combination (stimulant plus non-stimulant) and stimulant-only therapy on 1 year incident cardiovascular events using TMLE, reporting the average treatment effect as a risk difference (ATE = E[Y_{A=1}] – E[Y_{A=0}]). We used a Super Learner ensemble to predict both the cardiovascular outcome regression and the treatment mechanism (propensity score), and then applied TMLE’s targeting step to refine this prediction and obtain doubly robust, efficient effect estimates with 95% confidence intervals.

RESULT: The analytic cohort included 550 patients, of whom 83 received combination therapy and 467 received stimulant-only therapy. Within 1 year, 26 patients experienced the cardiovascular composite outcome. TMLE estimated an average treatment effect of -0.0141, suggesting that combination therapy was associated with an approximately 1.4% lower adjusted risk 1-year risk than stimulant-only therapy. The 95% CI was [-0.0437, 0.0155], which crosses 0, indicating no statistical difference at 0.05 level.

CONCLUSION: After adjusting for baseline confounding, we found no statistically clear cardiovascular risk difference between the two medication classes.

Ongoing work will strengthen confounder control by incorporating discharge note-derived variables, including surgery history and ADHD severity, extracted through LLM fine-tuning with GRPO on synthetic clinical notes and then applying the fine-tune weights to the original discharge notes for severity classification. These notes derived variables will be added to the causal model, and TMLE will be re0fit to assess whether richer clinical meaningfully changes the current findings.

10:50 AM - 11:10 AM: Bohan Zhou

The Signed Wasserstein Barycenter Problem

Abstract: Recent years have seen growing interest in the deep connections between optimization and sampling. In particular, Langevin dynamics for sampling can be interpreted as the gradient flow of the relative entropy in the space of probability distributions. A natural question is whether such connections can be extended to discrete state spaces. 

Building on the new interpretation of MCMC as a gradient flow with respect to the graphical Wasserstein metric, we propose a class of Nesterov-type algorithms to accelerate MCMC sampling on graphs. The corresponding continuous-time formulation can be viewed as a damped Hamiltonian flow in probability space. We establish theoretical results on convergence and acceleration for some user-specified setting, and present numerical examples demonstrating improved accuracy and convergence speed of sampling on multimodal distributions and real datasets.

11:10 AM - 11:30 AM: Rongyi Dai

Stochastic Compressible Euler Equations with Frictional Damping: Existence of L∞ Martingale Solutions and Asymptotic Porous Medium-Like Behavior

Abstract: We study the one-dimensional isentropic compressible Euler equations with linear (frictional) damping, subject to multiplicative, white-in-time stochastic forcing. The system is posed on a bounded interval with L∞ initial data and Dirichlet boundary conditions imposed on the momentum. We establish the global-in-time existence of L∞ martingale solutions that satisfy an appropriate entropy inequality. Then, we analyze the long-time behavior of these solutions and show that, under suitable assumptions on the noise, they converge almost surely and exponentially fast to a constant steady state of the system. The limiting density is well-approximated by the asymptotic solution of the deterministic porous medium equation, while the momentum exhibits the asymptotic behavior predicted by Darcy's law.

11:30 AM - 11:50 AM: Evan Davis

Studying Self-Organized Criticality in a Mass Shedding Stochastic Process

Abstract: We investigate self-organized criticality in a mass shedding stochastic process model built from the following simple rules: 1) at every unit of time, a mass clump is shed from the origin with i.i.d. mass amount, 2) clumps move at a speed that is a power law in their mass, and 3) when two clumps collide, they form a larger clump with the mass summed. Numerical experiments show that these simple rules lead to self-organized criticality. The scaling relation of the self-similarity can also be characterized via a heuristic argument. Aiming toward a rigorous understanding of this phenomenon, we derive the associated Kolmogorov forward equation describing the evolution of the distribution of the mass, resulting in an integro-differential equation. We show that this equation admits a family of self-similar stationary solutions.

11:50 AM - 12:10 PM: Satish Chandran

Score-Free Sampling via Energetic Variational Inference, Poisson-Nernst-Planck Dynamics, and Yukawa Potentials

Abstract: We present PNP-EVI, a particle-based sampling framework that extends the energetic variational inference (EVI) method to allow for score-free sampling. The method is rooted in the classical Poisson-Nernst-Planck (PNP) drift–diffusion model from electrolyte theory, where sampling particles are treated as positive charges attracted toward a fixed negative charge background encoding the target through reference samples alone. The free energy functional couples an entropic exploration term, computed via kernel density estimation, with an electrostatic interaction that drives particles toward the target. An implicit Euler discretization of the resulting particle dynamics yields a proximal minimization problem with guaranteed monotone energy decrease. We also investigate the use of Yukawa-type kernels for sample generation in higher dimensions for particle-based variational inference (ParVI) tasks.

10:50 AM - 11:10 AM: Xue Feng

Learn to Evolve: self-supervised Neural JKO Operator for Wasserstein Gradient Flow

Abstract: The Jordan-Kinderlehrer-Otto (JKO) scheme provides a stable variational framework for computing Wasserstein gradient flows, but its practical use is often limited by the high computational cost of repeatedly solving the JKO subproblems. We propose a self-supervised approach for learning a JKO solution operator without requiring numerical solutions of any JKO trajectories. The learned operator maps an input density directly to the minimizer of the corresponding JKO subproblem, and can be iteratively applied to efficiently generate the gradient-flow evolution. A key challenge is that only a number of initial densities are typically available for training. To address this, we introduce a Learn-to-Evolve algorithm that jointly learns the JKO operator and its induced trajectories by alternating between trajectory generation and operator updates. As training progresses, the generated data increasingly approximates true JKO trajectories. Meanwhile, this Learn-to-Evolve strategy serves as a natural form of data augmentation, significantly enhancing the generalization ability of the learned operator. Numerical experiments demonstrate the accuracy, stability, and robustness of the proposed method across various choices of energies and initial conditions. I’ll also sketch extensions to some other applications and close with open discussion

11:10 AM - 11:30 AM: Mingsong Yan

Infinite-Node Limits of Continuous-Depth Graph Neural Networks

Abstract: Continuous-depth graph neural networks, also known as Graph Neural Differential Equations (GNDEs), extend Graph Neural Networks (GNNs) with the continuous-depth architecture of Neural ODEs. A natural question is how GNDEs behave as the size of the underlying graph grows. In this talk, we introduce Graphon Neural Differential Equations (Graphon-NDEs) as the infinite-node limit of GNDEs and establish their well-posedness. We prove trajectory-wise convergence of solutions of GNDEs to solutions of Graphon-NDEs as the number of nodes tends to infinity, and we provide explicit convergence rates for GNDEs on both deterministic and random graphs. These results offer theoretical justification for a practical and computationally efficient strategy: deploying GNDE models trained on smaller graphs to much larger ones. Numerical experiments further support our theoretical findings.

11:30 AM - 11:50 AM: Natanael Alpay

On the Complex SGD with Application to Kernel Regression in Reproducing Kernel Hilbert Spaces and Directional Bias.

Abstract: Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large scale convex optimization problems due to its simple implementation and scalability. Some objectives, such as those found in complex neural networks, benefit from Gradient Descent-like update with a newly defined gradient that allows for complex parameters. This complex variant of the Gradient Descent method has been proposed, but convergence guarantees have not yet been provided. We propose a variant of SGD that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to Gradient Descent as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for Kernel Regression problems. We provide empirical results demonstrating the efficacy of the complex SGD in Kernel Regression problems utilizing complex reproducing kernel Hilbert spaces. In particular, we demonstrate we may recover superoscillation functions and Blaschke products from the Fock Space and Hardy Space, respectively.

Joint work with Emeric Battaglia

11:50 AM - 12:10 PM: Jack Luong

Self-similar Imploding Solutions of 1D Compressible Euler Equations with a Far Field Cutoff

Abstract: Smooth imploding solutions to the symmetric, isentropic, compressible Euler equations have been well-studied, inspired by the work of Guderley. However, these smooth imploding solutions are shown to be numerically unstable and difficult to compute in practice. On the other hand, the imploding solution of Kidder has a closed form solution and is numerically computable but is also unbounded in the far field. We consider Kidder's formulation in one dimension in which the unbounded far field condition is replaced with a constant density cutoff of the initial data. Strikingly, a non-centered rarefaction emerges from the cutoff and suppresses the implosion. We present an exact analytic solution to the problem with the cutoff, supported by numerical simulations.

10:50 AM - 11:10 AM: Scott Little

Geometric Langlands of Seiberg-Witten Elliptic Koopman D2-Tori

Abstract: Seiberg-Witten theory is a supersymmetric (SUSY) Yang Mills N=2-gauge theory developed by Seiberg and Witten in the early 1990s with real world solutions. For many configurations, an elliptic curve used to define the Seiberg-Witten 4D moduli space manifold. The correlation between elliptic curves and moduli space is a basis for the Langlands program. There are numerous applications for Seiberg-Witten theory, from quantum field theory, string theory, quantum computing and engineering, fluid dynamics, and energy propulsion systems.

The Langlands program, originally conjectured by Robert Langland in the 1960’s, has developed over the years into what many refer to as the ‘Rosetta stone’ and ‘grand unified theory of mathematics’. Historical research includes work by Srinivasa Ramanujan on modular forms of prime factors. The Langlands program has touched upon most major fields in mathematics and attempts to explain the relationship between these fields. There has been much progress, including solutions for finite fields by Drinfeld and Laumon in the 1980s, and in solving certain instances of modular forms by Frenkel, Gaitsgory and Vilonen. Possibly the most well-known solution is Andrew Wiles' proof of Fermat's Last Theorem, the modularity theorem for elliptic curves over the rational numbers.

The linear Koopman Operator Theory includes a state space of infinite dimensions to control a finite dimensional nonlinear dynamic system. Stochastic string theory is referred to as “postmodern” string theory. The strings are treated not as discrete objects but as probabilistic spaces to account for quantum uncertainties and nonlinear effects. 

The first and second papers in this series relate Anti-de-Sitter Spacetime Conformal Field Theory Correspondence or AdS/CFT Duality to Feynman-Kac stochastic string solutions in Mellin Transform Space. The third paper correlated Stochastic Feynman-Kac AdS/CFT solution to the Boltzmann Machine. The fourth paper was a KOT KvN Integral coupled to the previous Feynman-Kac stochastic string solutions. The fifth paper is a correlation between the Koopman Operator to the AdS/CFT Boltzmann Machine mapped to KAM tori 2D-branes of Dirac-Born-Infeld field string action. The sixth paper correlated elliptic curves to Koopman D2-DBI brane KAM Tori on complex space. 

In this paper, we take the results from previous papers into the Langlands program using stochastic Koopman D2-DBI brane elliptic curves, defined as Seiberg-Witten curves, and the corresponding harmonic analysis KAM tori modular forms. The theorem and proof are based on previous theorems and demonstrate continuity between the subject matter in the series. 

Additional applications are quantum gravity stochastic strings, fluid dynamics, cloaking electromagnetic black holes, cosmic strings gravity waves, gamma rays pulsars, machine learning-AI neural networks Boltzmann Machines, Feynman-Kac path integral, Schrödinger equation, Black Shoals economics finance, holographic AdS/CFT, chaos fractals, Koopman non-linear chaotic complexity.

11:10 AM - 11:30 AM: Yifan Gu

Transient Synamics of Bidisperse Neutrally Buoyant Thin-Film Flow

Abstract: We investigate the transient dynamics of the pressure-driven channel flow of bidisperse, neutrally buoyant, spherical, and non-Brownian particles. Two continuum models, diffusive flux and suspension balance, are considered. We apply the lubrication approximation to their governing equations with fast and equilibrium timescales. We then obtain simplified transient nonlinear partial differential equations and steady-state ordinary differential equations, respectively. We numerically solve the equations by a semi-implicit-explicit Runge-Kutta method and a nonlinear shooting method, respectively. Numerical simulations then reveal that the vast majority of equilibration times over a wide range of parameters significantly exceed the flow timescale, even by a factor of 200. This directly contrasts the corresponding equilibration times in monodisperse flow, which are on par with the flow timescale. There is also as much as an 80% difference between early-time transient and steady-state particle distributions. We also compare our results with available experimental data and find that, in more than 80% of the cases, transient solutions achieve smaller mean squared errors than the steady-state counterparts. Our work thus highlights the importance of practically meaningful timescales, as idealized long-term limits can obscure transient but physically observable effects.

11:30 AM - 11:50 AM: Lingyun Ding

Beyond Hele-Shaw: Systematic Reduced Modeling of Flow and Scalar Dynamics in Thin-Gap Microfluidics

Abstract: Thin-gap geometries are common in microfluidic devices, arising naturally from the constraints of fabrication techniques. However, simulating the resulting flows in three dimensions is computationally intensive, and classical 2D approximations—such as the Hele-Shaw model—fail to capture essential physics, including inertial effects and near-wall dynamics. We present a systematic approach for deriving accurate two-dimensional (2D) equations that approximate the full three-dimensional (3D) Navier–Stokes equations and their coupled scalar counterparts. The method follows the framework of the Method of Weighted Residuals, with physically informed choices of basis and weight functions guided by asymptotic analysis of the relevant flow quantities in the thin-gap limit. We validate the resulting 2D model against full 3D simulations across a range of representative microfluidic and inertial flow configurations, including coaxial flow devices, Dean flows, micromixers, and centrifuge-on-a-chip geometries. Our results show that the model maintains high fidelity even beyond the classical Hele-Shaw regime and provides a robust foundation for higher-order corrections and multiphysics extensions, such as electrokinetic transport and magnetohydrodynamic flows. This work enables efficient, accurate simulation of thin-gap flows and offers immediate utility in the design and optimization of microfluidic systems.

11:50 AM - 12:10 PM: Hongyi Guan

CuPyMag: GPU-accelerated FEM Solver for Multiphysics Micromagnetics

Abstract: We present CuPyMag, an open-source Python framework designed for large-scale, multiphysics micromagnetic simulations on GPUs. CuPyMag solves governing PDEs using finite elements within a GPU-resident workflow, in which key operations, such as right-hand-side assembly, spatial derivatives, and volume averages, are tensorized via CuPy’s BLAS-accelerated backend. Moreover, CuPyMag uses the Gauss-Seidel projection method for the time integration of the nonlinear Landau-Lifshitz-Gilbert (LLG) equation. This approach ensures numerical stability for larger time steps than typical explicit methods and reduces the nonlinear dynamics into a sequence of well-conditioned Poisson-type equations, ensuring high efficiency on GPUs. Benchmark results demonstrate that CuPyMag achieves sub-linear scaling for overall runtime, and benefits significantly from the high memory bandwidth of modern GPUs. Overall, our work highlights how proper mathematical structures that exploit the high-throughput memory hierarchy of modern GPUs can accelerate computationally intensive, coupled PDE systems in materials science.

1. Lorenzo Collier: "Conditioning Elections: The Instability with Dynamic Voters"

Abstract: The stability of electoral outcomes is a cornerstone of democratic legitimacy, yet real-world elections increasingly exhibit volatility driven by dynamic voter populations and shifting preferences. By treating elections as a function f(x) where x represents voter choice, we can study the conditioning of election outcome functions under perturbations, analogous to sensitivity analysis of numerical solvers. This paper investigates the mechanisms underlying instability in elections when voters and their beliefs evolve over time, focusing on the effects of conditioning decisions on anticipated future changes. By surveying classical and contemporary models of voting rules under dynamic settings, this paper demonstrates how seemingly stable aggregation mechanisms can lose robustness in the face of preference fluctuations, and analyze the strategic implications when voters or election designers incorporate information about forthcoming population changes into their choices. Through mathematical analysis and illustrative simulations, it is revealed that conditioning on future voter dynamics introduces new avenues for equilibrium selection, strategic manipulation, and unpredictability. Using data from current and historical datasets from the American National Election Survey, along with the most recent Cooperative Election Study dataset, we find that factors such as quality of campaigns and perceived fairness of elections are major factors in stabilizing or destabilizing elections. These findings highlight critical vulnerabilities and design principles for election systems in dynamic societies, offering both theoretical insights and practical considerations for sustaining stability in democratic processes.

2. Jonathan Tran: "Using Optimal Transport Aligned Latent Embeddings for Fluid Flow Analysis"

Abstract: As fluid flows live in infinite-dimensional (or high-dimensional in the discretized case) spaces, it is advantageous for many downstream tasks to represent flow fields in a low-dimensional manifold. The “manifold hypothesis” postulates that high-dimensional datasets often live on some lower-dimensional submanifold. Methods such as principal component analysis (also often referred to as proper orthogonal decomposition) or multidimensional scaling provide linear submanifold representations of data that often have intuitive interpretations. Recently, however, neural-network-based autoencoder methods have grown in popularity due to their superior compression performance. In principle, autoencoders aim to learn a latent space submanifold that is homeomorphic to the manifold on which the data lies. However, if an optimal network exists, it is clear that any diffeomorphism of the latent space is an equally optimal latent representation, meaning that it is difficult to interpret the geometry of the latent space outside of qualitative observations. Here, we combine autoencoders with multidimensional scaling to train an autoencoder that also attempts to align Euclidean distances in the latent space with distances computed using unbalanced optimal transport. We apply this optimal transport-based analysis to separated flows past a NACA 0012 airfoil with periodic heat flux actuation at the leading edge. The cases considered are at a chord-based Reynolds number of 23,000 and a free-stream Mach number of 0.3. For each angle of attack, optimal transport-based embedding succinctly captures the different effective regimes of flow responses and control performance, characterised by the degree of suppression of the separation bubble and secondary effects from laminarisation and trailing-edge separation.

3. Ryan Anderson: "A Quantifier-Free Description of the Semi-Algebraic Set of Value Functions in Reinforcement Learning"

Abstract: We characterize the set of feasible value functions in infinite-horizon partially observable Markov decision processes (POMDPs) with memoryless stochastic policies. Our main contribution is a novel, quantifier-free description of the feasible value functions for a POMDP, which form a semi-algebraic set. We construct the semi-algebraic set of value functions in terms of finitely many inequalities which are polynomial in the transition, observability, and reward parameters of the POMDP via Cramer's rule. Our description recovers previous results for both fully observable and partially observable Markov decision processes. Finally, we use this novel characterization to discuss differences in optimization dynamics between fully observable and partially observable MDPs.

4. Bob Zhao: "Modeling Cellular State Transitions: An RNA Velocity–Guided Neural Stochastic Framework for Forward Simulation"

Abstract: Single-cell RNA sequencing (scRNA-seq) can represent each cell as a high-dimensional gene expression vector and enables computation of RNA velocity, which estimates short-term directional change. However, existing methods primarily infer fate probabilities

rather than simulate explicit future trajectories. In this project, I model cellular transitions as a stochastic process in gene expression space and develop an RNA velocity–guided stochastic simulation framework to simulate forward evolution. Transitions are proposed using local similarity and velocity alignment, and accepted via a Metropolis–Hastings inspired criterion. Terminal states are treated as absorbing states. This approach generates explicit simulated trajectories from a single snapshot, shifting scRNA-seq analysis from static inference toward dynamic prediction of cellular evolution.

5. Alexandru Vajiac: "Identifying Optimal Graph Structure for Business Operational Models"

Abstract: Identifying optimal graph structures is essential for modeling complex dependencies in business operations and enabling causal reasoning. This study formulates business processes as directed acyclic graphs (DAGs) and applies causal structure learning techniques to infer relationships from observational data. We employ a combination of score-based methods (e.g. Bayesian Information Criterion optimization, Bayesian Dirichlet equivalent uniform) and constraint-based approaches (e.g. conditional independence testing, arc strength) to estimate candidate graph structures. The framework integrates domain constraints to reduce equivalence classes and improve identifiability. Empirical results show that the learned graph structures enable more accurate estimation of intervention effects compared to traditional regression-based models. This approach supports counterfactual analysis and provides actionable insights for optimizing operational strategies, resource allocation, and system performance.

6. Ahmed Kaffel: "Modeling Unsaturated Flow in Thin Multi-Layered Media made of Absorbent Swelling Porous Materials"

Abstract: Wicking in thin media plays a crucial role in liquid absorption across a wide range of applications, including wipes, diapers, medical devices, sportswear, filtration, batteries, and oil spill remediation. This chapter presents a mathematical modeling framework for analyzing liquid absorption and solid deformation during unsaturated two-phase flow in thin, swelling porous media under isothermal conditions. Using a volume-averaging approach, three-dimensional pointwise mass balance equations are reduced to quasi two-dimensional averaged equations. These macroscopic balance laws are coupled with appropriate constitutive relationships. A closure model is introduced to describe inter-layer liquid mass exchange, while a deformation model based on nonlinear elasticity accounts for layer compression.

The resulting framework significantly improves computational efficiency, enabling faster and more cost-effective simulations of absorbency processes in partially saturated porous media such as fiber–hydrogel composites. These models provide enhanced insight into wicking dynamics in thin media and support optimization across diverse industrial and environmental applications.

7. Liora Mayats Alpay: "Causal Graph Network Learning with an Extended Front-Door Criterion for Interventional Effect Estimation"

Abstract: Estimating the effect of an intervention from observational data is an important problem and challenging in causal inference. This becomes much harder when there are hidden confounding factors and when the relationships between variables are complex and nonlinear. In this work, we introduce a new way to use graph networks for cause-and-effect analysis, building on an improved front-door approach to estimate the effect of interventions. Our method uses graphs to look at how different factors, treatments, steps in the process, and results are linked. The new version of the front-door approach lets us use more types of graph structures than the original method. Our graph-based method helps identify the complex patterns in large datasets. This way, the model keeps the clear cause-and-effect meaning of graph methods while also using the power of modern machine learning. We explain what we assume for our method to work, describe how we estimate effects, and show how using graphs can make it easier to study the effects of interventions in complex situations where data was not collected in an experiment. This work helps bring together cause-and-effect theory and graph-based machine learning in a way that is more flexible and useful.

8. Surendra Maharjan: "Mapping of African Thirstwaves Reveals Clustered Hotspots and Large-Scale Modulation of Atmospheric Water Demand Extremes"

Abstract: Hydroclimatic risk assessments have traditionally emphasized precipitation deficits, yet under rapid warming, drought impacts are increasingly amplified by surges in atmospheric evaporative demand. Here we provide a continent-scale characterization of “thirstwaves”—multi-day episodes of anomalously high reference evapotranspiration (ET₀)—across Africa using daily ERA5 reanalysis (1980–2023) and the FAO-56 Penman–Monteith formulation. We map spatial patterns of thirstwave frequency, duration, and intensity, and show strong geographic contrasts that align with Africa’s seasonal timing of peak ET₀, reflecting the migration of insolation and circulation regimes. Thirstwave frequency is elevated across North Africa, the Sahel, the Horn of Africa, and parts of southern Africa, while humid equatorial regions exhibit weaker demand extremes. Hotspot analysis reveals clustered, non-uniform concentrations of recurrent thirstwaves in East and Central Africa, West Africa, southern Africa, and Madagascar, highlighting transition zones where land–atmosphere coupling and seasonal moisture limitations heighten vulnerability. Using cross-wavelet analysis, we further demonstrate that large-scale climate variability modulates thirstwave occurrence, with the most persistent shared variability concentrated at interannual periods and most consistently associated with ENSO-related indices, while other modes contribute episodically in a region- and timescale-dependent manner. Finally, a two-period change analysis indicates that increases in thirstwave frequency are most strongly linked to rising vapor pressure deficit, warming, enhanced winds, and declining soil moisture, whereas changes in duration and intensity show weaker linear associations with individual drivers. Together, these results establish a physically grounded, event-based framework for diagnosing demand-driven extremes and identifying African regions where increasing atmospheric “thirst” may intensify risks to ecosystems, agriculture, and water security.

9. Yaw Acquah: "Math Misperception"

Abstract: This research investigates how network structure and homophily shape individuals’ perceptions of majority opinion within social systems. Using simulated networks, we analyze how local interactions can lead to systematic misperceptions of global support. We developed and implemented a simulation framework in Python utilizing tools such as NetworkX, NumPy, and Pandas to model opinion dynamics across multiple graph structures, including path, cycle, random, and small-world networks. A two-community stochastic block model was employed to parameterize homophily, enabling controlled variation in the likelihood of within-group versus between-group connections.

Within this framework, each agent forms beliefs about overall support based solely on their immediate neighbors. Misperception is quantified as the difference between the true proportion of supporters and the average perceived proportion across nodes. Through extensive simulations across varying levels of homophily and network sizes, we conducted statistical and visual analyses to identify key trends. The results reveal that increasing homophily consistently amplifies negative misperception (i.e perceived support exceeds true support), leading individuals to underestimate true majority support. This effect persists across network types, though its magnitude varies with structural properties.

This work highlights the critical role of network topology in shaping collective perception and demonstrates how local information constraints can produce widespread belief distortions. The findings contribute to a deeper understanding of opinion formation and have implications for studying polarization and information diffusion in real-world social networks.

10. David Culver: "Using Sinousodal Neural Networks to create Implicit Representations of Nearshore Ocean Waves from Irregularly Spaced LiDAR Data"

Abstract: We use a class of neural networks known as sinusoidal representation networks or SIREN’s to create accurate representations of LiDAR point clouds of nearshore ocean waves. We collected the data by flying a gas-electric drone equipped with a LiDAR sensor above extreme swell breaking off China Rock in Monterey. The LiDAR sensor simultaneously projects multiple LiDAR scans perpendicular to the crests of the incoming waves, resolving the sea surface. The SIRENs then use the LiDAR data to train on, optimizing themselves to become functions that output sea surface elevation as functions of space and time. We then tested the reliability of these networks by training multiple networks over the same data and examining the differences between the networks and the error from the LiDAR returns. Due to the functional construction of neural networks, the approximated sea surface is a continuous and fully differentiable function in space and time, allowing estimation of sea surface derivatives.

11. Ishan Jha: "Maximum Entropy Regularization for Gradient Pathology Mitigation in Physics-Informed Neural PDE Solvers"

Abstract: In the past decade, deep learning-based methods have found significant application in the approximation of solutions to partial differential equations (PDEs). Physics-informed neural networks (PINNs) encode physical constraints by embedding the PDE residual into the training loss, yielding mesh-free differentiable surrogates. However, PINNs exhibit documented gradient pathologies in which loss-function gradient dynamics become stiff, producing solutions that minimize the collocation residual without meaningfully satisfying the global physics of the system. We demonstrate empirically that these pathologies extend beyond the loss function, and that the gradient distribution of the predicted solution in poorly-performing PINNs exhibits sharp concentration near zero, reflecting a failure to resolve the spatial complexity of the true solution. To address this, we introduce Gradient Entropy Regularization (GER), which augments the PINN loss to include the Gaussian entropy of the predicted solution's gradient distribution. The Gaussian surrogate is justified by the Principle of Maximum Entropy, which states that among all distributions consistent with given moment constraints, the Gaussian uniquely maximizes entropy, yielding differentiable lower bound on the true gradient entropy. GER achieves a statistically significant reduction in relative $L^2$ error across benchmark problems, providing concrete validation of entropy-based regularization as a principled tool in neural network-based numerical analysis.

12. David Licerio: "Probabilistic ICU Readmission Risk Modeling Using Bayesian Networks"

Abstract: Unplanned ICU readmission results in high costs for individuals' health as well as high penalties for hospitals due to re-admissions. The aim of this paper is to develop an interpret-able model to identify conditional readmission risk for various patient profiles. Using the MIMIC database a causal model was engineered using Bayesian Network. The variables used in the model included demographics like age and gender, socioeconomic variables and admission type and location. Continuous variables were transformed into categorical as is necessary for this type of model. Readmission probabilities were calculated by isolating individual variables while holding others constant. Readmission risk varied across demographic socioeconomic and admission characteristics. Older patients, as well as minorities and men, tended to have higher readmission probabilities compared to younger, white and female patients. Individuals with private insurance tended to have better outcomes than those on Medicare. The study provides insight into how individual background variables like demographics and socioeconomic status can have on the life of a patient.

13. Gabriel Alpay: "Baseline Mortality Prediction in Lung Cancer: An Interpretable Machine Learning Approach Using MIMIC-IV"

Abstract: This study evaluates a machine learning pipeline to predict mortality in lung cancer patients using the MIMIC-IV database. A cohort of 4,220 patients was identified using ICD-9 and ICD-10 diagnostic codes. To establish a predictive baseline, a demographics-only XGBoost model (evaluating age, gender, race, and insurance) was tested, yielding an average Area Under the Receiver Operating Characteristic Curve (AUC-ROC) of 0.547 (+- 0.026) using 10-fold cross-validation. A full clinical model was then developed by integrating derived first-day intensive care unit (ICU) features, including the Charlson Comorbidity Index (CCI), SOFA scores, mechanical ventilation status, metastatic status, and laboratory values. Using automated class-weight balancing, the full XGBoost classifier significantly improved predictive performance to an average AUC-ROC of 0.694 (+- 0.031) under 10-fold cross-validation. SHAP (Shapley Additive Explanations) analysis demonstrated that baseline health burden (CCI) and cancer severity (metastatic status) were the primary predictors of mortality, alongside physiological distress signals such as SOFA scores and hemoglobin levels. This comparative approach mathematically validates the use of standardized diagnostic and clinical ICU data for automated risk assessment in cancer-related critical care.

14. Cameron Hajaliloo: "Structure-Aware Content Classification on Social Networks via Diffusion Graph Learning and LLMs"

Abstract: Understanding how information diffuses through social networks is essential for mitigating public health risks, particularly when misleading claims can rapidly influence behavior. We present a hybrid framework that integrates diffusionbased graph learning with large language models (LLMs) to analyze content patterns in online information ecosystems. We construct an interaction graph linking posts through replies, quotes, and hashtag overlap, and apply the Diffusion State Distance (DSD) metric to embed this sparse network into a fully connected structural space that captures long-range influence. This diffusion-based representation reduces hub bias and reveals latent connectivity patterns overlooked by traditional graph metrics. A human-LLM labeling pipeline extracts and verifies factual claims, enabling robust misinformation detection even in highly imbalanced datasets. Our results show that DSD enhances sensitivity to subtle misinformation diffusion and scales efficiently to large datasets. By uncovering hidden structural pathways through which harmful content spreads, our framework offers practical tools for real-time monitoring and mitigation of online misinformation, contributing to safer and more trustworthy information environments.

15. Ruoxi Zhao: "Autonomous AI Agents for Sequential Decision-Making: Optimization and Evaluation in Finance and Scientific Domains"

Abstract: Autonomous AI agents powered by large language models (LLMs) hold immense potential for solving complex sequential decision problems under uncertainty, yet their optimization and evaluation remain challenging. This work introduces a modular Agent Engine framework that equips LLMs with stateful tools, memory systems, and human-in-the-loop safeguards for safe, adaptive decision-making. We evaluate the framework in two diverse environments: (1) financial trading using Backtrader and OpenBB, where agents execute algorithmic strategies and are benchmarked via automatically generated multiple-choice questions (MCQs) with verifiable ground truth; and (2) scientific optimization, focusing on inverse problems in dynamical systems (e.g., ODEs/SDEs) and black-box optimization tasks. Our methodology formulates agent training as a Markov Decision Process (MDP) amenable to reinforcement learning (RL), enabling systematic performance comparisons against tools like GitHub Copilot. Preliminary results demonstrate the framework's efficacy in generating domain-specific tasks and evaluating agent adaptability.

16. Jenna Coffman: "Perception Dynamics"

Abstract: This paper proposes a model that describes how individuals' perceptions change over time through social interactions. Using a stochastic block model with two groups, supporters and opponents, each node’s label is updated based on its neighbors until the labels converge to a single consensus value. Under the specified update rule, this final consensus value represents the ideal level of support for a given issue. This work extends prior research on stochastic block models by shifting focus from explaining the underestimation of support for policies to modeling how people’s beliefs may be influenced by those around them.

17. Bohan Zhou: "Accelerating MCMC Algorithms on Finite Graphs"

Abstract: Recent years have seen growing interest in the deep connections between optimization and sampling. In particular, Langevin dynamics for sampling can be interpreted as the gradient flow of the relative entropy in the space of probability distributions. A natural question is whether such connections can be extended to discrete state spaces. Building on the new interpretation of MCMC as a gradient flow with respect to the graphical Wasserstein metric, we propose a class of Nesterov-type algorithms to accelerate MCMC sampling on graphs. The corresponding continuous-time formulation can be viewed as a damped Hamiltonian flow in probability space. We establish theoretical results on convergence and acceleration for some user-specified setting, and present numerical examples demonstrating improved accuracy and convergence speed of sampling on multimodal distributions and real datasets.

Abstract: Learning solution operators for nonlinear PDEs governing fluid dynamics is challenging due to high dimensionality, multiscale structure, and limited observations. Multi-operator learning and PDE foundation models aim to address this by pretraining on a wide range of spatiotemporal datasets, leading to zero-shot and few-shot generalization across varying parameter regimes. In this talk, we survey recent architectures for PDE foundation models, including operator-based approaches and autoregressive formulations for spatiotemporal prediction. We will focus on applications to incompressible and compressible flows, with an emphasis on accuracy and generalization. We will highlight some open problems and promising directions for future research. 

Abstract: One of the most important aspects of mathematical modeling is the ability to examine counterfactual circumstances. Social science and public policy are fields where counterfactual analysis can be especially important.  In this presentation, we focus on one particular social and policy challenge: the so-called crisis of reproducibility.  We aim to show that mathematical simulation modeling can provide insights into interventions proposed to improve reproducibility in life and social science research.

Experimental design and statistical data analysis are thought to be major contributors to reproducibility problems, particularly practices such as P-hacking to produce results below this threshold, selective reporting of positive studies, and designing studies with too few subjects.  A number of interventions have been proposed, ranging from insisting on culture change to registered reporting to reducing the threshold of significance. In this talk, we use an evolutionary agent-based model comprised of researchers who test hypotheses and strive to increase their publication rates.  Properties like effort, effect size, number of hypotheses tested, and registered reporting varied across researchers, with evolution driven by total publication value.  We see that reducing the signficance threshold does not necessarily reduce P-hacking but that it may improve the false positive rate in the literature.  Likewise, registered reporting shows promise in reducing false positives but may require changes in the academic incentive system for widespread adoption.

3:20 PM - 3:40 PM: Samrat Mondal

Complex Role of Superspreaders: Faster Spread, Weaker Selection, Faster Evolution

Abstract: Heterogeneity in contact networks is known to accelerate the spread of infectious diseases through the presence of superspreaders, but its evolutionary consequences remain less understood. Here we study how network heterogeneity shapes the fate of competing pathogen strains in a stochastic susceptible–infected–susceptible framework. We show that heterogeneous networks act as strong suppressors of selection: Both advantageous and disadvantageous mutants exhibit fixation probabilities close to neutral expectations, in stark contrast to well-mixed populations. We derive an analytical theory that captures this effect through a single suppression factor determined by network structure and infection dynamics, and validate it against simulations on synthetic and empirical contact networks. Mechanistically, suppression arises because most transmission events are effectively neutral, while selection acts only in rare transmissions. As a consequence, heterogeneous networks substantially increase the persistence of deleterious mutants, elevate mutation–selection balance, and accelerate multi-step evolutionary processes such as fitness valley crossing, specially in rare mutation limit. Our results reveal a fundamental trade-off induced by superspreaders: while they enhance epidemic spread, they weaken selective pressures and thereby promote evolutionary diversification.

3:40 PM - 4:00 PM: Chase Brown

Contamination Dynamics of Active Rods in Long Micro-Channels

Abstract: Active rods are elongated, self-propelled particles that have emerged as an effective model for many motile micro-organisms, such as bacteria and sperm cells. When confined to a microfluidic channel, active rods exhibit a marked feature in their motility: they swim against the fluid flow (positive rheotaxis) along channel walls. Following this motility pattern, bacteria can utilize the walls and corners of medical catheter tubing to swim upstream and contaminate the device, possibly resulting in upstream swimming-induced infection. An individual trajectory of such a particle typically involves alternating between intervals of upstream swimming at the channel boundary, and downstream motion in the fluid bulk. In this talk, I will present the active rod model confined by a long three-dimensional cylindrical micro-channel of various cross-section geometries. The model successfully captures both positive rheotaxis and alternating down- and upstream swimmings. The focus of the talk will be on the dynamics of the active rod concentration profile along the major axis of the microchannel, describing the spatial spread of a population of swimming microorganisms under confinement. Using the active rod model, we show that both upstream and downstream swimming times are exponentially distributed. Based on these results, we find the concentration profile by introducing a one-dimensional continuous-in-time random walk model, a generalization of 1D Brownian motion, for active rods, and by deriving the corresponding partial differential equation governing concentration. This equation can be viewed as an extension of the telegrapher’s equation. Both analytical and numerical results for this reduced model show good agreement with the original three-dimensional active rod simulations, as well as with recent experimental observations of bacterial contamination in thin microchannels.

4:00 PM - 4:20 PM: Navaira Sherwani

Integrative 3D Multiscale Modeling of Drosophila Wing Disc Eversion

Abstract: Understanding the biochemical and mechanical cues that cause epithelial sheets to reorganize themselves into three-dimensional organs remains a central challenge in biology. During Drosophila wing-disc eversion, a pseudostratified pouch

folds outward and fuses into a bilayer that goes on to form the adult wing. This project unites quantitative experiments with a 3D multiscale mechano-chemical (M3D) model to decode the late-stage morphogenesis of the Drosophila wing imaginal disc during eversion. The work couples: (i) a GPU-accelerated particle-based model that resolves apical, basal and extracellular-matrix mechanics on a deforming epithelial surface; (ii) reaction–diffusion models for hormone (ecdysone) and morphogen signaling (Dpp, Wg) extended down to intracellular Rho1/Cdc42 dynamics; and (iii) machine-learning pipelines—Gaussian-process surrogate modeling, Bayesian optimization and neural-network solvers—to calibrate and accelerate simulations against time-lapse light-sheet imaging, biomechanical perturbations and quantitative immunostaining. Iterative experimentation will map how spatially patterned actomyosin contractility, cell-ECM adhesion and ECM stiffness drive coordinated cell reshaping, layer coupling and tissue folding. The resulting framework will yield predictive, systems-level insights into how hormonal timing interfaces with morphogen gradients to orchestrate organ-scale shape changes.

4:20 PM - 4:40 PM: Roberto Ceja

Robustness of Network Inference Algorithms under Network Class Misspecification

Abstract: Inferring phylogenies in the presence of hybridization remains a difficult problem. As a result, many current methods for reconstructing phylogenetic networks are restricted to a simple class of networks known as level-1. This restriction arises from theoretical considerations rather than empirical evidence, with real data possibly arising from complex networks.

In this work, we evaluate the robustness of two level-1 network inference methods, SNaQ and NANUQ+, through a simulation study, when the input data originates from a more complex network. Specifically, we investigate whether these methods can accurately recover important features of the true species network, such as the circular order (the arrangement of taxa around a network), taxa of hybrid origin  (taxa that arose from a hybridization event), and other structural properties.

Our results show that while both methods are accurate in recovering the circular order of the network, they lack in other network features such as determining which taxa are a product of a hybridization event. These results highlight the strengths and limitations of current level-1 methods under model misspecification, guiding the interpretation of their outputs in practice.

3:20 PM - 3:40 PM: Vinh Nguyen

Mean Field Limits for Flocking Models with Nonlinear Velocity Alignment

Abstract: We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment.  The mean-field limit is proved in two settings: deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic models in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation.

3:40 PM - 4:00 PM: Yi Fu

Fragility in a Togashi–Kaneko Stochastic Model with Mutations

Abstract: A prototypical mathematical model of autocatalysis is the Togashi–Kaneko (TK) stochastic model, which is known for displaying dramatic switching between dominant species. In this work, we study the TK model with added mutations. We establish a rigorous stochastic averaging principle that describes slow dynamics in terms of certain ergodic means of fast variables. Beginning with two species, we demonstrate an extreme sensitivity to small asymmetries in autocatalytic reaction rate constants. Even very slight departures from symmetry can lead to qualitatively different dynamical behaviour. We refer to this striking sensitivity as “fragility”. We also explore extensions to more complex networks, finding that similar effects may occur for four species but appear absent for three. These preliminary explorations for multiple species point to a wealth of open questions for future research.

4:00 PM - 4:20 PM: Siting Liu

Icon to Genicon: Probabilistic Operator Learning with Uncertainty Quantification

Abstract: In-context operator networks (ICON) learn solution operators for ODEs/PDEs by conditioning on example initial/boundary data and their solutions. I’ll present a probabilistic interpretation: ICON implicitly computes the posterior predictive mean given the context. Using random differential equations, this connects ICON to Bayesian inference and naturally leads to a generative variant, GenICON, that samples from the posterior predictive for principled uncertainty quantification. The framework unifies operator learning under a Bayesian lens and delivers uncertainty-aware predictions.

3:20 PM - 3:40 PM: Xinyue (Tracy) Yu

MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data

Abstract: Collective behaviors that emerge from interactions are fundamental to numerous biological systems. To learn such interacting forces from observations, we introduce a measure-valued neural network that infers measure-dependent interaction (drift) terms directly from particle-trajectory observations. The proposed architecture generalizes standard neural networks to operate on probability measures by learning cylindrical features, using an embedding network that produces scalable distribution-to-vector representations. On the theory side, we establish well-posedness of the resulting dynamics and prove propagation-of-chaos for the associated interacting-particle system. We further show universal approximation and quantitative approximation rates under a low-dimensional measure-dependence assumption. Numerical experiments on first and second order systems, including deterministic and stochastic Motsch-Tadmor dynamics, two-dimensional attraction-repulsion aggregation, Cucker-Smale dynamics, and a hierarchical multi-group system, demonstrate accurate prediction and strong out-of-distribution generalization.

3:40 PM - 4:00 PM: Yasamin Jalalian

Learning Differential Equations from Scarce Data: A Kernel-Based Approach

Abstract: In many problems in computational science and engineering, observational data are available while the underlying physical models remain unknown. Learning such models from data is a central goal in scientific machine learning, yet many existing approaches require numerous solution examples, dense measurements, or computationally expensive training procedures. In this talk, I present Kernel Equation Learning (KEqL), a kernel-based framework for learning differential equations and their associated solution maps in scarce-data regimes, both when the number of training solutions is small and when each solution is observed at only a limited number of locations.

KEqL formulates equation learning within an RKHS/Gaussian process-based framework that simultaneously reconstructs the unknown solution and identifies the functional form of the differential equation. I will describe the practical KEqL methodology and discuss the accompanying error analysis, which establishes quantitative worst-case bounds for the learned equation under suitable assumptions.

Finally, I will present numerical benchmarks spanning ODEs and PDEs, including variable-coefficient problems, demonstrating improved robustness, computational efficiency, and significant accuracy gains over state-of-the-art equation learning methods. I will also discuss connections between this framework and operator learning and inverse problems, and outline future directions, both theoretical and computational.

4:00 PM - 4:20 PM: Liyao Lyu

Learning and Data Assimilation McKean-Vlasov Dynamics from Particle Data

Abstract: Learning interaction laws in collective systems and incorporating partial observations into mean-field dynamics are two closely related challenges. We propose a unified framework that addresses both. First, we introduce a measure-valued neural network (MVNN) that learns measure-dependent interaction (drift) terms directly from particle trajectories, using scalable distribution-to-vector embeddings. We establish well-posedness, propagation of chaos, and universal approximation with quantitative rates under a low-dimensional structure.

Second, we develop Multiscale Nudge, a data assimilation method that injects macroscopic observations into microscopic dynamics via a Wasserstein gradient-based nudging term with a tractable particle implementation. Under suitable assumptions, we prove exponential decay of the $L^2$ error up to a model-misspecification bias.

Experiments on kinetic, collective-motion, and chaotic systems demonstrate accurate interaction learning, improved forecast performance, and strong generalization from incomplete observations.

4:20 PM - 4:40 PM: Martin Hernandez

Model-Free Continuous-Time RL for Mean-Field Control

Abstract: We propose a continuous-time framework for infinite-horizon mean-field control in the context of reinforcement learning. On the model-based side, we characterize the problem through the mean-field HJB equation, introduce the associated q-functional, and derive a policy gradient formula that supports policy iteration. We then turn to the unknown-dynamics setting, where only short-time snapshots are available, and show that a mean-field PhiBE approximation yields a model-free counterpart that approximates the optimal policy. The result is a unified perspective linking mean-field control, continuous-time reinforcement learning, and data-driven approximation.

3:20 PM - 3:40 PM: Qijin Shi

No-Arbitrage with Rough Paths: Rigidity, Kreps–Yan, and Unbiased Rough Integrators

Abstract: Rough path theory provides a natural pathwise framework for continuous non-semimartingale models, extending classical stochastic calculus beyond the Itô setting. We study the extent to which this framework can support frictionless no-arbitrage markets, formulated via a No Controlled Free Lunch (NCFL) condition. We establish a rough Kreps–Yan theorem, proving that NCFL is equivalent to the existence of an equivalent measure under which the price driver is an unbiased rough integrator. Our main contribution is a classification of such unbiased rough integrators for progressively richer classes of controlled portfolios, building on a combinatorial description of one-dimensional rough paths. For Markovian-type portfolios, the admissible drivers are time-changed Gaussian-Hermite rough paths. Furthermore, allowing path-dependent signature-type portfolios further narrows the admissible class to the Itô rough path lift of Brownian motion. Thus, sufficiently rich frictionless rough-path market models are forced back toward the classical semimartingale paradigm. The framework applies to one-dimensional α-Hölder rough paths for arbitrarily small α>0 within the tensor algebra setting.

3:40 PM - 4:00 PM: Gerves Francois BANIAKINA

Mathematical Analysis of Regime Transitions in Hedge Funds Performance using Markov Chain Theory

Abstract: This paper develops a mathematical framework for analyzing hedge fund performance regimes using  nite-state, time-homogeneous Markov chains. Monthly returns from the EDHEC Hedge Fund Index are discretized into three economic regimes—crisis, normal, and growth—based on quantile thresholds. Using this discretized state space, the empirical transition matrix is constructed and examined to characterize regime dynamics. Stationary distributions, eigenvalues, and convergence properties are computed to evaluate long-run behavior and regime persistence. Visualization tools, including heat maps and Markov chain diagrams, highlight the probabilistic structure of transitions. Results show that hedge fund regimes exhibit strong persistence, particularly within normal-return states, and that eigenvalues close to one imply slow convergence toward the stationary distribution. These  ndings demonstrate that even simple  nite-state Markov Chain models can capture key structural features of hedge fund performance dynamics while remaining analytically tractable and appropriate for discrete-mathematics–based analysis.

4:00 PM - 4:20 PM: Ka Lok Lam

Feynman Formula for Discrete-time Quantum Walks

Abstract: A discrete-time quantum walk is a unitary model of a quantum particle evolving on discrete time and space. In this talk, we show that it can be represented as a Feynman–Kac-type functional of a classical four-state Markov additive process with complex potentials. This representation allows us to connect the spectral properties of the stochastic process to the asymptotic distribution of the discrete-time quantum walk. We further provide a probabilistic proof that the discrete-time quantum walk scales to the linear Dirac equation in continuous time and space, yielding a stochastic representation of the linear Dirac equation and naturally leading to Monte Carlo solvers for these PDEs. This talk is based on joint work with Jean-Pierre Fouque and Tomoyuki Ichiba (UCSB).

4:20 PM - 4:40 PM: Georg Menz

Data-driven Breakpoint Detection with the QML Estimator

Abstract: We study quasi-maximum likelihood (QML) breakpoint estimation for covariance regime switches in multivariate time series. We move beyond the classical framework and show that regime switches can be detected as soon as the signal-to-noise ratio is high enough. We identify a quantitative global recovery threshold that compares signal separation between regimes to signal fluctuations within regimes, and show its sharpness via an explicit counterexample. We also discuss further developments in breakpoint detection in high dimensions.

3:20 PM - 3:40 PM: Florian Wolf

Global Solutions to Non-Convex Functional Constrained Problems with Hidden Convexity

Abstract: Constrained non-convex optimization is generally intractable, yet many problems in control and reinforcement learning possess hidden convexity — they admit a reformulation as a convex program via an unknown nonlinear transformation. Since this transformation is typically inaccessible, we study algorithms that operate directly in the original space using only (sub-)gradient oracles.

We develop the first algorithms with provable global convergence guarantees for this setting, covering both smooth and non-smooth regimes. Our results require no constraint qualifications, handle equality constraints, and match the complexity of unconstrained hidden convex optimization — despite operating in a non-convex landscape.

3:40 PM - 4:00 PM: Therese Landry

Towards a Mathematical Theory of the Optimal Transport of Sediment

Abstract: Via the Monge-Kantorovich theory, we prove the existence of unique global weak solutions to equations describing the sediment flow in the evolution of fluvial land surfaces, with constant water depth.  A solution to this PDE determines a land surface.  These PDEs describe the so-called transport-limited situation, where all the sediment can be transported away given enough water. This is in distinction to the detachment-limited situation where we must wait for rock to weather (to sediment) before it can be transported away.  The Wasserstein metric framework will permit access to computational techniques from optimal transport theory.  Time permitting, we may discuss existing numerical techniques involving fractal interpolation due to Birnir and Cattan, as well as numerical schemes which efficiently approximate gradient flows for Wasserstein metrics.  Some synthesis of the two methods could produce substantial improvements in computing time for solutions to our sediment flow PDE. This is joint work with Bjorn Birnir.

4:00 PM - 4:20 PM: Jacob Murri

Continuous Data Assimilation for State Reconstruction in Mixed Convection from Temperature-Only Measurements

Abstract: Many important problems in engineering and science require reconstructing full flow states using only partial observations. Inspired by the Charney conjecture, which states that temperature observations alone are sufficient to recover all other state variables in simple atmospheric models, we explore a continuous data assimilation algorithm called nudging (or Newtonian relaxation) which uses continuous-in-time data from temperature sensors to reconstruct the full temperature and velocity state of a system modeled by the Boussinesq equations which exhibits both forced and natural convection. The nudging algorithm requires simulating the evolution of a set of assimilating state variables governed by equations utilizing a feedback control term that pushes the temperature state towards the available sensor measurements using finite-element interpolation. We characterize the physical regimes (in terms of the Rayleigh and Reynolds numbers) where this procedure succeeds in synchronizing the assimilating state variables with the true states, and analyze decay rates in terms of the coefficient on the feedback control term. We also compare the performance of nudging with other continuous data assimilation algorithms on this problem.

4:20 PM - 4:40 PM: Justin Marks

Constructing Odd-Sized Preference Profiles in the Stable Marriage Problem with Exponential Growth in the Number of Stable Matchings

Abstract: In the Nobel Prize-awarded stable marriage problem, n men and n women rank participants of the opposite gender in order of preference.  The goal of our work is to find preference profiles that maximize the number of stable matchings, which is a long-standing research problem.  Each participant has n! ways of ranking the opposite gender yielding n!^{2n} possible preference profiles. A method is developed for constructing odd-order preference profiles in the stable marriage problem with exponential growth in the number of stable matchings.  This method effectively extends a known method for even orders to be applicable to all orders.  Several of the odd-order profiles establish new records for the number of stable matchings. A formula is also developed.

3:20 PM - 3:40 PM: Truong Vu

Dynamics of Polynomial Root Sets Under Iterated Differentiations

Abstract: 

3:40 PM - 4:00 PM:  Mingyu Liu

Finding Minimal Energy Paths: Algorithms, Convergence, and Applications to Infinite-Dimensional Patterned Landscapes

Abstract: We present numerical studies of minimal energy paths (MEPs) in complex energy landscapes. First we prove the convergence of the climbing string method that locates saddle points and provide numerical validation of the convergence. We then develop numerical methods for computing local minimizers and the MEPS between them governed by a strain-gradient energy functional. Our computations reveal structured transition pathways and an energy landscape exhibiting clear pattern selection, demonstrating that the MEP approach can be used to explore transitions of stable states in infinite-dimensional spaces.

4:00 PM - 4:20 PM: Yutong Ren

PhiBE Difference Learning

Abstract: We study online and off-policy learning in continuous-time reinforcement learning (CTRL), where the system dynamics follow an unknown stochastic differential equation(SDE) but only discrete trajectory data are available. Building on the recently proposed Physics-informed Bellman Equation (PhiBE), we develop a model-free learning framework that enables incremental policy evaluation and Q-learning in continuous time. The proposed algorithms support online updates and off-policy data while avoiding model-based PDE formulations. We establish convergence guarantees and convergence rates under linear function approximation by exploiting the ellipticity of the infinitesimal generator of the underlying SDE.

4:20 PM - 4:40 PM: Yuming Zhang

Discretization Error Arising from RL Approximations to Continuous-Time Optimal Control

Abstract: While reinforcement learning (RL) typically employs discrete-time Markov Decision Processes (MDPs), its connection to continuous-time optimal control remains a significant theoretical challenge. This work bridges this gap by investigating a class of relaxed control problems with uncontrolled diffusion coefficients. We establish explicit convergence rates for optimal feedback controls across discrete, continuous, relaxed, and classical regimes. If time permits, I will also discuss the convergence properties of the policy iteration algorithm within this framework. These findings provide a rigorous theoretical foundation for implementing RL in stochastic, continuous-time environments.

3:20 PM - 3:40 PM: Edith Zhang

Reaction--Diffusion Equations on Graphons

Abstract: In this talk, I will begin by introducing graphons, which are infinite-size limits of adjacency matrices of sequences of growing graphs. I will then define graph reaction-diffusion (RD) equations, which are systems of differential equations that are

defined on the nodes of a graph. For a sequence of growing graphs that converges to a graphon, the solutions of the sequence of graph RD equations also converge. The limiting solution solves a nonlocal differential equation that we call a graphon RD

equation. Furthermore, the graph RD equation is related to a stochastic birth-death process on graphs. I will show that this birth-death process converges to the graphon RD equation via a hydrodynamic limit.

3:40 PM - 4:00 PM: Daniel Gurevich

Weak-Form Discovery of Interpretable Tensor Models

Abstract: The homogeneity and isotropy of physical space require mathematical models of natural phenomena to be equivariant with respect to symmetries including translations, rotations, and reflections. Unstructured data-driven model inference leads to an intractable optimization problem due to the high dimensionality of the spatiotemporal data and the model search space. While popular approaches to dealing with symmetries such as data augmentation and symmetry-constrained neural network architectures may promote equivariance, they make only limited impact on tractability. On the other hand, symbolic algorithms can enforce strong inductive biases that not only incorporate known symmetries but also leverage symbolic representations that can parsimoniously parametrize a wide range of models. This talk introduces such an algorithm, which exploits tensor representations of the symmetry group to model systems via sets of partial differential or integro-differential equations represented as tensor networks. We have taken inspiration from programming language theory to design and implement formal languages representing models within these search spaces. The use of canonical forms and a satisfiability modulo theory solver allows for efficient automated synthesis, evaluation, and symbolic deduction of these models, with coefficients in each equation identified using a homogeneous variant of weak-form sparse regression. In conclusion, several applications to physical systems will be discussed.

4:00 PM - 4:20 PM: Raymond Stefani

Analyzing Significant Olympic Improvements in Winning Performances, Women’s Training and Efficiency Equalizing with Men, Building Blocks of Improvement and Reducing PED Use

Abstract: Winning times were analyzed for all seven men’s and seven women’s 2024 Olympic swimming races also contested at half the distance. All seven male and female winners from 2026 swam twice as far at the same velocity as their half-distance counterparts two generations earlier, a feat that would have been considered unachievable two generations before. For swimming, rowing, speed skating and running, using kinesiology and physics, if women have achieved equal training and efficiency with men, the current velocity ratio of female champions/male champions would be calculable from their relative lean to weight ratios which was found to now be true, therefore women have become equally trained and efficient, compared to male champions. Plots of cumulative improvements for male champions and for female champions in running and swimming were created over Olympic history. Consistent with the Olympic year of each introduction by the World Anti-Doping Agency of methods to find and reduce use of Performance Enhancing Drugs (PEDs), was future reduction in the quality of winning performances, indicating the success of those anti-PED actions.

4:20 PM - 4:40 PM: Jimmie Adriazola

Proximal Spectral Coordinates for Model Reduction in Nonlinear Dynamics

Abstract: We present a model reduction framework for nonlinear dynamical systems that uses state-dependent spectral coordinates built from an operator associated with the evolving state. Unlike fixed reduced bases, these coordinates adapt to the local structure of the dynamics, allowing reduced models to remain informative in strongly nonlinear regimes. This talk will introduce the idea through the lens of integrable Hamiltonian mechanics where the spectral and evolution operators collapse to the classical Lax pair, and show how the framework provides an operator-based, geometry-aware approach to reduced-order modeling for nonlinear PDEs.