Dr. Jay Schweig 

Department Head 

Associate Professor

Dept. of Mathematics

Oklahoma State University 

Abstract: Geometric optimization problems for Laplacian eigenvalues have a rich history, dating back to Rayleigh’s 19th-century work. A cornerstone result is the Faber-Krahn Theorem, which states that among all open subsets of IRn with fixed volume, the ball minimizes the first Dirichlet eigenvalue. This principle has inspired numerous extensions, involving higher eigenvalues, alternative boundary conditions, additional constraints, and more general operators. Beyond their theoretical appeal, these problems have a plethora of applications in physical problems.

In this talk, I will provide an overview of this active research area, highlighting key results and open questions. I will also present recent joint work with S. Snelson on optimal shapes for the first Dirichlet eigenvalue of an elliptic operator with irregular coefficients—a problem of genuine applied interest that poses significant challenges due to low regularity of solutions.

Abstract: Research on fluid dynamic computation has been very active and of interest to both mathematicians and engineers. The PDEs raised in modeling compressible flow, such as the compressible Navier–Stokes (NS) equations are fundamental in gas dynamics with various applications in numerous important areas. Preserving the positivity of density and internal energy without losing conservation is crucial to stabilize the numerical simulation. However, constructing a high-order accurate conservative and positive-preserving scheme with large time step size in the sense of standard hyperbolic CFL remains a challenging task. In this talk, we propose a fully discrete semi-implicit scheme for solving the compressible NS equations within the Strang splitting framework. Our scheme preserves conservation of arbitrarily high order in space. The positive-preserving property for up to Q3 space discretization is achieved by using the matrix monotonicity. For Qk (k ≥ 4), we combine discontinuous Galerkin methods with large-scale non-smooth optimization techniques to construct a limiter that ensures bound preservation without losing conservation and accuracy. Such a constrained optimization can be efficiently solved by the generalized Douglas–Rachford splitting method. Our optimization method is very efficient in terms of minimal memory requirements and low iteration costs. It scales well for each iteration, with a complexity of O(N) where N is the total number of mesh cells. Our scheme for solving compressible NS equations enjoys the standard hyperbolic CFL on time step size. Numerical experiments suggest that our scheme produces satisfactory non-oscillatory solutions when physical diffusion is accurately resolved. It is well-suited for simulating realistic physical and engineering problems.

Abstract: In this work, we build upon the numerical scheme from [R. Adhikari, I. Kim, Y. Lee, and D. Sheen, \textit{Numer. Methods Partial Differ. Equ.}, 80 (2024)] to achieve improved accuracy in approximating the flux variable in second-order elliptic boundary value problems (BVPs). A considerably better flux approximation is obtained by iterating the variational equation on a fixed mesh---incorporating the computed flux into the right-hand side data. The accuracy improvement is more pronounced for smaller values of the user-defined parameter $\delta \in (0,1]$ in the equation. An analysis of the role of $\delta$ and the effect of iteration in enhancing accuracy is presented. The proposed iterative scheme motivates an efficient adaptive finite element method (FEM), where the solution from a coarser mesh is included in the right-hand side terms for solving the variational equation on the next finer mesh. An adaptive finite element algorithm is developed based on this scheme and the error estimator first introduced in [A. Alonso, \textit{Numer. Math.}, 74 (1996), pp. 385–395]. Furthermore, a fully computable error bound for the computed flux is derived using the abstract result of [M. Vohralik, \textit{Math. Comp.}, 79 (2010), pp. 2001-2032 ]. Numerical experiments are included to confirm the theoretical findings.

Abstract: Predicting protein-ligand binding affinity is a fundamental challenge in drug discovery. Recent advances in deep learning have led to the development of numerous models, many of which rely on three-dimensional protein-ligand complex structures and focus primarily on affinity prediction. In this study, we introduce an Algebraic Graph Neural Network (AGNN) model designed to encode molecular structures into a low-dimensional graph representation while preserving critical biochemical interactions. While algebraic graph theory has been widely used in physical modeling and molecular studies, traditional methods often struggle to accurately capture the complexity of biomolecular interactions. To address this limitation, our proposed AGNN model leverages multiscale weighted colored subgraphs to describe molecular interactions through graph neural network. These representations allow the model to effectively learn the geometric and topological features of protein-ligand complexes. The AGNN model integrates graph convolutional layers and attention mechanisms to refine feature extraction and improve the interpretability of learned embeddings. Furthermore, we incorporate gradient boosting decision trees (GBDTs) to enhance the prediction of binding affinities by capturing nonlinear relationships between molecular features. Our approach is extensively validated using benchmark datasets, including PDBBind and CASF-2016, demonstrating superior performance in binding affinity prediction compared to state-of-the-art scoring functions. 

Abstract: Front-tracking is an adaptive numerical approach that explicitly tracks the interface between distinct mediums as a hypersurface moving through a rectangular grid. The method provides sharp resolution of the wavefront and prevents unwanted mixing between neighboring cells of different materials, at a computational cost. In high resolution simulations, adaptive mesh refinement can reduce this expense by only refining near the interface, while keeping a coarser grid in the smooth regions. To this end, the front-tracking based software library FronTier is coupled with the block-structured, adaptive mesh refinement library AMReX to provide high-fidelity simulations of the Rayleigh Taylor instability using multiple levels of refinement. Scaling of the application is explored, and simulations are validated against well-known results. 

Abstract: Artificial intelligence (AI) has ushered in a new era of scientific discovery, yet its direct application to biomolecular data remains challenging due to intricate structural complexity, exceedingly high dimensionality, pronounced nonlinearity, and multiscale characteristics. To tackle these issues, we integrate cutting-edge mathematical frameworks—including persistent homology, persistent Laplacians, and evolutionary de Rham–Hodge methods—into AI pipelines. Persistent homology, a cornerstone of topological data analysis (TDA), reduces high-dimensional biomolecular structures into low-dimensional topological descriptors, while our persistent Laplacians detect non-topological shape variations via non-harmonic spectra. For volumetric data (e.g., protein electron densities), we formulate evolutionary de Rham–Hodge models and boundary-induced graph Laplacians to capture multiscale features and reduce computational costs. By interfacing these mathematical tools with advanced machine learning strategies—such as ensemble learning, manifold learning, graph neural networks, and transformers—we reveal key molecular mechanisms in SARS-CoV-2 transmission and evolution, specifically explaining how infectivity strengthens and antibody resistance emerges. These interdisciplinary methods highlight the power of combining sophisticated geometry and topology with AI to analyze complex biological systems.

Finally, we extend our mathematical–AI framework to partial differential equation (PDE) approaches in biophysical modeling. Building on the shape representations and complexity reductions introduced earlier, we focus on the Poisson–Boltzmann equation for electrostatic interactions within biomolecular systems, showing how AI-driven numerical solvers and graph-aided symbolic learning can tackle large-scale problems. This synergistic approach—merging topological simplifications, PDE-based modeling, and advanced AI—underscores how modern mathematics elevates AI-driven predictions in the biosciences, paving the way for new frontiers in understanding molecular interactions and designing effective therapeutics.

Abstract: Homogeneous solutions play an important role in the study of fluid equations. In this talk, I will discuss homogeneous steady states of 3D incompressible Navier-Stokes equations (NSE) with an isolated singularity or finite singular rays.

I will first talk about the Landau solutions, which are a family of explicit (-1)-homogeneous axisymmetric no-swirl solutions with one singularity at the origin, discovered by Landau in 1944. In 1998 Tian and Xin proved that all (-1)-homogeneous axisymmetric solutions with one singularity are Landau solutions. In 2006 Sverak proved that all (-1)-homogeneous solutions smooth on the unit sphere must be Landau solutions. I will then talk about some recent study on (-1)-homogeneous steady states of NSE with singular rays. I will describe the existence and classification of such solutions that are axisymmetric with two vertical singular rays. We classify all such solutions with no swirl and then obtain existence of nonzero swirl solutions through perturbation methods. I will then discuss some isolated singularity behavior of homogeneous solutions to Navier-Stokes equations, and present some removable singularity result. I will also establish the asymptotic stability for some of the axisymmetric no-swirl solutions we obtained, and talk about some anisotropic Caffarelli-Kohn-Nirenberg type inequalities we derived and applied in the study.

Abstract: Human mobility is natural on this connected planet and is crucial in infectious disease's temporal and spatial spread. To study the specific role of mobility levels, we equipped a metapopulation epidemic model with cellphone mobility data to capture daily population flows among counties within states. This approach allows for the incorporation of time-varying population movement, forming dynamic networks that are aggregated into a metapopulation framework. This, in turn, enables the development of a hybrid data-driven mechanistic model and facilitates realistic synthetic simulations of disease spread. The synthetic simulations of this hybrid model uncovered heterogeneous epidemic patterns and the geo-temporal (spatial and temporal) spread of the diseases that a traditional gravity model or fully connected networks cannot capture. This hybrid approach also captured complex transmission dynamics driven by real-world movement patterns, significantly improving traditional metapopulation models. The analysis was performed on all US states at the county level of daily population flow data to find how mobility flows affect the disease progression in the network. The primary findings of this research revealed that human mobility within the network and transmission rate at the heterogeneous county (higher transmission rate) could potentially “accelerate or flatten the curve” of infected dynamics. Noteably, we proposed a rigorous definition for hotspot or high-risk counties and superspreaders of disease spread across fifty US states. We validate the hotspots by comparing them with network centrality measures and population distribution data. Moreover, by aggregating countywide flows in all fifty states in the USA, we demonstrated that early monitoring of the disease spread in the metapopulation and reducing population flows from mobility hotspots can slow the epidemic and localize the outbreaks.

Abstract : Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of interface instabilities between two fluids. In this talk, we compare the Z and monotonicity preserving bounds variants of the high order Weighted Essentially non Oscillatory (WENO) methods using the front tracking method. First , three one dimensional test problems show the strengths and weaknesses of each method. Then the mach 1.21 simulations of Richtmyer Meshkov instability of air and sf6 gas are presented.

Abstract: We have nice invariants for compact topological spaces like singular homology and cohomology, and Betti numbers. In the case that your topological space is non compact, then these invariants could be huge or non even well defined.  L2-invariants generalize the classical invariants for non-compact spaces. I will give a review of them and some applications. 

Dr. Xu Zhang

Associate Professor

Dept. of Mathematics

Oklahoma State University