Mathematics &
Statistics Seminars

High-energy neutrino flavor transitions provide a sensitive probe of physics beyond the Standard Model, particularly in regimes where ultra-high-energy cross-sections and extended mixing structures become relevant. In this work, we model neutrino flavor evolution using the PMNS framework and higher-order transition indices to generalize the standard oscillation probability structure. These calculations incorporate cosmic-ray-scale interaction cross-sections, where neutrino-electron scattering reaches approximately 1.691 × 10⁻³³ cm² for E_ν ∼ 1000 GeV.

We then explore conditions under which neutrino states may mix with supersymmetric neutralinos—fermionic combinations of photino, zino, and higgsino fields—commonly proposed as WIMP dark matter candidates. In R-parity-violating SUSY scenarios, modified neutral fermion mass matrices and non-zero sneutrino vacuum expectation values can enable neutrino-neutralino mixing, allowing oscillations into heavier neutralino-like states over long baselines or extreme energies. Representative density transitions are computed to illustrate how such extended mixing could influence flavor populations and potentially connect neutrino oscillation phenomenology with dark matter signatures.

These results highlight a mathematically consistent pathway for linking neutrino flavor physics with supersymmetric dark matter models and motivate further investigation into observational consequences for cosmic neutrino experiments and next-generation dark matter searches.

We present an operator-learning framework on a data-driven multiscale space for heterogeneous multiscale problems. The local multiscale space is constructed from fine-grid snapshots using local data compression, providing a compact representation that preserves localized heterogeneous features. A neural operator is then trained to learn the map from permeability fields to multiscale solution coefficients, replacing the intrusive Galerkin solve with a non-intrusive data-driven prediction. To obtain reliable predictions, we evaluate the residual of the learned solution and apply a small number of localized correction iterations. This inexpensive localized post-processing step improves the solution, particularly in the energy norm, without introducing a costly physics-based residual loss. Numerical results are presented for nonlinear flow in random heterogeneous porous media.

In this presentation, I will begin by introducing shuffled regression and entropic optimal transport (EOT) as one possible tool for solving shuffled regression. A common approach for this optimization is to use a first-order optimizer, which requires the gradient of the OT distance. For faster convergence, one might also resort to a second-order optimizer, which additionally requires the Hessian. I will present the analytical derivatives of EOT, provide a brief overview of numerical condition numbers, and explain how to compute a crucial linear system robustly. Through analytical derivation and spectral analysis, we identify the numerical instability caused by the singularity and ill-posedness of a key linear system, prove the asymptotic limits of its condition number when both sample size N goes to infinity and regularization strength ε goes to 0, and improve the efficiency and robustness of computation. Finally, I would like to discuss future work as well as extensions.

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We present recent advances in computational and mathematical methods for analyzing data that lie on nonlinear spaces, with a focus on techniques grounded in Riemannian geometry. Two manifolds drive the discussion: the manifold of symmetric positive definite (SPD) matrices, which arises naturally in covariance-based image and signal descriptors, and the manifold of diffeomorphisms, which underpins large-deformation image registration.

A central object in both settings is the geodesic distance, whose evaluation requires solving boundary value problems that quickly become prohibitive in high dimensions. We discuss variational formulations for the associated registration and analysis problems, and introduce a new method for k-means clustering on high-dimensional, non-Euclidean manifolds based on Fréchet maps. The idea is to embed manifold-valued data into a low-dimensional Euclidean space via distances to a small set of reference points, chosen to preserve the geometric information relevant for clustering. Once embedded, standard Euclidean algorithms apply, yielding substantial reductions in runtime over intrinsic methods while maintaining clustering accuracy.

Taken together, these results illustrate how Riemannian principles can be made computationally tractable for classification, clustering, and registration of high-dimensional manifold-valued data.

This is joint work with Nicolas Charon, Demetrio Labate, Robert Azencott, and collaborators.

By employing new extended multilevel hierarchy construction principles, AMG can be applied to many new types of problems. The extended principles include general rules for choosing relaxation, constructing the coarse-level variables using principal component analysis, the coarse-to-fine interpolation using weighted least squares, coarse-level equations derived variationally, and a quantitative performance predictor of the multi-level cycle convergence rate, called the mock cycle. As an illustrative example, a fast solver and eigensolver for the highly indefinite 1D and 2D Helmholtz equation are developed.

This is a joint work with Achi Brandt, Karsten Kahl and Oren Livne.

We study the distribution on the nonnegative orthant of the hypersphere arising by transforming with the absolute value a von Mises--Fisher distribution. The positive von Mises--Fisher distribution has a tractable density form with closed-form normalizing constant. Several of its properties, including likelihood inference, moments, and mode, are derived. A new distribution on the simplex is proposed as a byproduct. Numerical experiments illustrate the properties of the model, while an application in text mining and a synthetic example demonstrates the benefits of the new distribution.

Living conditions, income, trauma, discrimination, and social support are examples of sociocultural factors that have a substantial impact on health outcomes, especially in opioid abuse recovery. Clinical narratives provide richer information regarding these factors than traditional surveys, which frequently have a narrow scope and self-report bias. This study uses natural language processing (NLP), transformer models, and LLM-augmented labeling for identifying sociocultural factors of mental health from EHR clinical notes and combines NLP-based detection with causal inference for better understanding their effects on outcomes such as opioid use disorder (OUD) recovery. We introduce a Human-in-the-Loop-LLM Interaction for Annotation (HLLIA) framework and a Multilevel Hierarchical Clinical-Longformer (MHCL) model to classify sociocultural factors of mental health from discharge summaries. These determinants are then analyzed using a Siamese neural network-based subgroup discovery approach to estimate causal effects on Opioid use disorder progression. The framework is extended to Breast Cancer Recurrence (BCR), where Clinical-Longformer multi-task classification extracts determinants from oncology notes and causal models such as SNN-SD, IPTW-RA, and CEVAE estimate Conditional Average Treatment Effects (CATEs), enabling identification of actionable drivers of disease outcomes and supporting personalized healthcare interventions.

In decision-making applications, such as parameter study, design optimization, optimal control, and inverse problems, we need to repeatedly solve forward problems. However, subject to the model complexity and the discretization fineness, it may take a long time to complete a single forward simulation with even advanced computing capabilities. Reduced order modeling (ROM) has nowadays become a popular and actively researched computational technique to reduce the computational cost of simulations while minimizing the error introduced in the reduction process.

In this talk, we will present recent ROM developments to several applications, including energetic material modeling, quantum molecular dynamics, compressible gas dynamics, and thermodynamics. Depending on the nature of the underlying PDEs and numerical simulations, we explored the design of different ROM techniques, including projection-based nonlinear model reduction, indicator-based and interpolation-based local reduced order models, and physics-consistent non-intrusive system identification. We will introduce our open-source software libROM, and its application to enable accelerated physical predictions.

This work was performed at Lawrence Livermore National Laboratory and partially funded by Laboratory Directed Research and Development (LDRD) Program by the U.S. Department of Energy (21-FS-042, 24-ERD-035). This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program through the LEADS SciDAC Institute under Project Number SCW1933 at Lawrence Livermore National Laboratory. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344.

Mixture models are widely used for modeling heterogeneous data populations. A standard approach is to assume that each mixture component follows a parametric kernel form, while model flexibility is achieved by using a large, possibly unbounded, number of such kernels. In many applications, however, parametric assumptions on latent subpopulation distributions may be unrealistic, motivating nonparametric modeling of the mixture components themselves. We study finite mixtures with nonparametric mixture components using a Bayesian nonparametric approach. We present conditions under which the individual component distributions are identifiable, and we establish posterior contraction behavior for the population density as well as for the densities of the latent mixture components. We also develop an efficient MCMC algorithm for posterior inference and demonstrate through simulation studies and real-world data examples that it is possible to learn complex latent subpopulation distributions efficiently. Theoretically, the posterior contraction rate for the component densities is nearly polynomial, representing a significant improvement over the logarithmic convergence rates that arise when estimating mixing measures via deconvolution.

Black hole accretion has to keep discharging angular momentum of its incoming matters, outwardly through the accretion disk, via turbulent torques. Searches for the source of this turbulence remain a hotspot of research. Hydrodynamic centrifugal instability, owing to its fundamental ubiquity, was at first hypothesized to be the source of turbulence in accretion disks. However, despite extensive experiments and simulations over the past 20 years that pushed Reynolds numbers (i.e., flow rotation speeds) up to orders of 10^5~10^6, no sustained hydrodynamic turbulence has been captured inside the "quasi-Keplerian", namely the most astrophysically relevant flow regime. [Note: This is intriguingly surprising as it seriously contradicts our existing knowledge that viscous flows, in general, can easily loss their stability to trigger turbulence/chaos under a little bit of finite (i.e., nonlinear) disturbances at merely a few hundred Reynolds numbers.]

Such a rare exceptional type of viscous flow led to a fundamental, yet long-standing open problem on nonlinear stability of quasi-Keplerian flows at high Reynolds numbers, which is utmost challenging to prove or disprove due to notorious nonlinearities of the fluid dynamics equations (thanks to Navier-Stokes). In this seminar, I will present my recent discovery of a viscous energy-Casimir identity that analytically implies, for the first time, global nonlinear stability throughout the entire quasi-Keplerian regime for infinitely large axisymmetric disturbances at arbitrarily high Reynolds numbers. In addition, the novel mathematical identity immediately reveals the exact physics underlying flow stability that perfectly matches numerical simulations. My theoretical analysis sets the first-principle cornerstone for nonlinear hydrodynamic stability of quasi-Keplerian flows and beyond.

We present a spatial-temporal eco-epidemiological model that integrates a Lotka–Volterra predator–prey system with a Susceptible–Infected–Susceptible (SIS) disease process and diffusion to capture spatial movement. The resulting reaction–diffusion system describes interactions among susceptible and infected prey and predator populations, incorporating ecological factors such as growth, mortality, predation, reproduction, and disease transmission. To investigate parameter-driven dynamics, we generate three large synthetic datasets via forward–Euler simulation (20k runs epidemic parameters varied), (20k, ecological parameters varied), and 100k, all parameters varied. Analysis reveals that epidemic variation prolongs transient dynamics, ecological variation promotes rapid convergence to disease-free states, and fully coupled variability yields the widest range of outcomes. Recovery rates most strongly influence infection levels, while no single parameter explains convergence time. Machine learning surrogates accurately predict equilibria and convergence, with best performance from MLP (R²≈0.95), Extra Trees (R²≈0.87), and Voting ensembles (R²≈0.81), demonstrating scalable emulation of eco-epidemiological dynamics without repeated simulation.

Medical imaging plays a critical role in disease diagnosis by providing non-invasive, detailed visualizations of internal body structures, aiding clinicians in detecting and assessing abnormalities. Accurate disease prognosis from medical imaging remains challenging due to the incompatibility of the prognosis model, the imbalances of data, and the complexity of model optimization. This study introduces a family of extended convolutional neural network methods that are systematically integrated with both advanced data balancing techniques and Bayesian optimization for hyperparameter tuning. These extended frameworks address, 1) class imbalance through three strategic resampling methods representing the cases under-sampling, oversampling, or hybrid; and 2) hyperparameter optimization by employing a Bayesian framework to efficiently navigate high-dimensional hyperparameter spaces. The performance of the extended frameworks is analyzed with three benchmark medical imaging datasets, namely, three-dimensional magnetic resonance imaging (MRI) scans of brain tumors, chest X-rays of pneumonia, and ultrasounds of breast cancers (or related soft-tissue lesions). The frameworks are evaluated and compared using three famous performance metrics, receiver operating characteristic, precision-recall, and classification accuracy, in which they demonstrate statistically significant improvements in prognostic accuracy.

This presentation explores a novel deep learning framework—ZICOM Poisson DNN—designed to model global lightning activity using count-based data. It begins by highlighting the importance of integrating machine learning techniques with statistical rigor, emphasizing the limitations of traditional approaches. The COM-Poisson distribution is introduced as a versatile alternative to the standard Poisson model, capable of handling both over- and under-dispersed data. The talk concludes with a comparison between predicted and observed lightning counts, demonstrating the enhanced performance and interpretability of the proposed method.

In the past three decades, the satellite observations and retrievals have been used in both monitoring the global severe weather and in climate research. At TAMUCC, databases of global precipitation events as objects from 2D or 3D fields are created with data from infrared and microwave radiometers and radars from geostationary and polar orbiting satellites since 1990s. These databases have served as a foundation to understand the physical processes of global cloud, precipitation, and convection in the past two decades. In this presentation, different types of observations and model products currently residing on TAMUCC servers are briefly described. The applications of using machine learning methods in analysis of these datasets are briefly introduced.

The fractional Korteweg–de Vries (fKdV) equation models the unidirectional propagation of weakly nonlinear dispersive waves of long-wavelength. The existence of periodic waves in the fKdV equation has been studied using fixed point, perturbative, and variational methods. From the point of view of variational method, the periodic waves are characterized as constrained minimizers of the energy functional subject to fixed mass and momentum.

In this talk, we are going to discuss two results regarding the existence and stability of the periodic waves of the fKdV equation. First, we are going to show the existence of positive periodic solution by proving that the Green’s function for the linear operator in the fKdV equation is strictly positive. Second, in the context of variational method, we are going to propose an alternative framework in which the periodic waves are constrained minimizers of the quadratic form of energy subject to fixed cubic part of energy and the zero mean. This new characterization allows us to fully unfold the region of existence of the periodic waves and to establish a sharp stability criteria based on the monotonicity of the map from the wave speed to the wave momentum

Reaction–diffusion systems remain an area of active interest due to their multifaceted applications and fundamental theoretical challenges. Despite being studied for over a century, a complete theory of well-posedness is still lacking. A central difficulty arises from the fact that certain mass-conserving systems (in the L^1 sense) may exhibit finite-time blow-up. Broadly, reaction–diffusion systems model the spatial dispersion of interacting species, biological, chemical, or physical where dispersal is governed by advection–diffusion operators or their variants, and interactions are described by nonlinear kinetic equations. The state variables often represent spatial densities such as concentration, population, or mass, with applications ranging from reactive flow, contaminant transport, combustion theory, and solid-state physics to population dynamics and the spatio-temporal spread of infectious diseases. We develop a mathematical model for the spread of a waterborne disease in a spatially distributed host population. The population occupies a bounded domain in the upper half-plane, with part of its boundary lying on the real axis. Infection occurs through a pathogen introduced via a waterway flowing along the boundary. This model illustrates how reaction–diffusion theory can capture complex dynamics of disease transmission in spatially heterogeneous environments, bridging theoretical advances with applications in epidemiology.

We present a novel numerical method for simulating quasi-static crack propagation in strain-limiting materials. Our approach utilizes a regularized variational model specifically formulated to overcome the challenge of crack-tip strain singularities. This is achieved through a logically consistent strain-energy density derived from nonlinear constitutive relationships. The resulting variational formulation is solved using an adaptive finite element method, guided by residual-based error estimates, to resolve features near the crack tip accurately. We provide convergence analysis and validate the algorithm's effectiveness with numerical examples.

This is a collaborative effort with my current postdoc, Dr. Ram Manohar. Additionally, this material is based on work supported by the NSF-DMS under Grant No. 2316905.

The integration of Reduced Order Modeling (ROM) and Deep Learning (DL) presents a promising avenue for enhancing computational efficiency and predictive accuracy. ROM techniques reduce complex systems into low-dimensional representations, preserving essential dynamics, while DL methodologies excel at learning intricate patterns from raw data. By combining ROM with DL, we aim to develop approaches that inherit merits from both disciplines. On one hand, we explore leveraging ROM as a preprocessing step to train DNNs with limited labeled data, addressing data scarcity. On the other hand, we intend to utilize DL to expedite ROM construction and learn ROMs directly from observational data. In this talk, we will elaborate on specific efforts made along this line.

In this talk, we discuss multigrid methods for the numerical solution of the Stokes-Darcy equations discretized by the Marker and Cell (MAC) scheme, a finite difference method. The resulting discretization leads to a double saddle-point system, and the design of fast solvers poses several challenges. We propose two block smoothers based on a block LDL decomposition of the matrix. To make them more computationally practical, we develop inexact versions based on damped Jacobi. Numerical experiences show that our approach is robust with respect to the meshsize and the physical parameters. This work is joint with Chen Greif from the University of British Columbia

We model dynamically changing candidate positions in the face of a dynamic electorate. We use Hegselmann-Krause (HK) opinion dynamics to model the changes in the electorate. The HK model can be reformulated from the probability density perspective. We analyze properties of the resulting PDE and look at their effect on the original particle model. Then candidates move along the gradient and are maximizing the votes. We use the combined candidate-voter model to demonstrate the possibility of discontinuous jumps in candidate behavior as parameters of the model are varied.

A direct perturbation method for studying dynamics of dark-bright solitons of the Manakov system in the presence of perturbations is presented. We combine multiscale expansion method, perturbed conservation laws, and a boundary layer approach, which breaks the problem into an inner region, where the bulk of soliton resides, and an outer region, which evolves independently of the soliton. We show that a shelf develops around the dark soliton component, with speed of the shelf proportional to the background intensity. Conservation laws of the Manakov system are used to determine the properties of the shelf and perturbed solutions. Our analytical predictions are corroborated by numerical simulations.

In this talk I will tell you about the Thomas-Fermi-Dirac-Von Weizsäcker (TFDW) theory in Quantum Physics, and about an Ohta-Kawasaki model inspired on problems in Material Science

In this talk, we will address an $hp$-adaptive discontinuous Galerkin finite element method (DGFEM) using both $h$ and $p$ refinement techniques to approximate the solution of a static crack boundary value problem. The mathematical model represents the behavior of a geometrically linear strain-limiting elastic body. We will discuss the existence of a unique discrete solution by applying Riesz's representation theory. Additionally, we address apriori error estimates for the DGFEM, expressed in both energy and $L^2$-norms. We showcase two numerical examples showing the proposed method's effectiveness in involving a manufactured solution and a domain featuring an edge crack.

This joint work with Dr. Mallikarjunaiah is supported by a grant from NSF-DMS Award No. 2316905.

The representation theory of logarithmic vertex operator algebras, especially for examples beyond rank one, has proven to be extremely difficult. One of the most interesting and well-studied families of logarithmic vertex operator algebras are the Feigin-Tipunin algebras. Their representation theory is especially interesting as it is related to the category of line operators for certain 3D quantum field theories. In this talk I will discuss how conjectures about the representation theory of Feigin-Tipunin algebras can be generated from braided categories constructed from the representation theory of quantum groups.

Sintering, a pivotal technology in additive manufacturing, transforms ceramic and metallic powders into solid objects. To achieve products with customized properties, a deep understanding of microstructure evolution during sintering is crucial. Our approach ensures thermodynamic consistency, deriving the driving force for particle motion from the system's free energy. As a result, our proposed phase-field-micromechanics model guarantees microstructure evolution that minimizes the system's energy. We rigorously validate this model against recent theoretical benchmarks. Subsequently, we employ it to simulate the microstructure evolution of polycrystalline powder particles, shedding light on the mechanisms governing crystallite growth.

The exploration and production of geothermal energy have been important missions for the energy contribution of the world, especially because geothermal energy is one environmentally friendly resource. There are plenty of geothermal reservoirs in the United States and China, but the subsurface situations are so complicated that it is hard to produce the geothermal resource economically. In this presentation, we summarized the current situations of the geothermal exploration in the United States and China of and studied the impact of injection wells on the geothermal production performance. Tracer tests were performed to test the connections between three injection wells and two production wells, and the Finite Difference Method and Streamline Algorithms based on Complex Analysis Potential are applied to simulate the temperature profile and flow trajectories and running time of the water from the injection well. The temperature evolution profile and the tracer test results were analyzed to get possible production feasibility and interconnection relations between different wells.

Data-driven anomaly detection methods typically model the expected behavior of the target system and score observations with respect to the expected norm. A threshold is invariably needed to identify data instances with high (or low) scores as anomalies. This presents a practical limitation on the applicability of such methods, since most modeling techniques are sensitive to extreme observations and the choice of the threshold. The issue is exacerbated in a streaming scenario, where the optimal thresholds vary with time.

This talk will explore the need for anomaly detection algorithms for time series and streaming datasets and their applications in predictive maintenance, fraud detection, public health, climate modeling, and risk management. We will also touch on the role of emerging AI/ML technologies, such as quantum and neuromorphic computing and foundation models, and discuss the potential challenges of implementing these technologies across different domains.