Virginia Tech Mathematics Colloquium – Fall 2026

Classical population models often rely on simple, unstructured differential equations that assume uniform life stages and instantaneous growth. In reality, species have stage-specific traits and behavioral adaptations that require more complicated mathematical tools. In this talk, I will illustrate how applied mathematicians collaborate with biologists to build mechanistic, first-principles models that capture multi-scale ecological dynamics. We begin with grazing, introducing co-limitation functions to capture how nutrient limitations alter feeding behavior. We then transition to growing up, using stage-structured and delay differential equations to model how high nutrient demands in juveniles create developmental bottlenecks. We demonstrate how stage-specific nutrient demands shift stability boundaries and control population oscillations under enrichment. Ultimately, we show how matching the right mathematical framework to biological reality uncovers novel dynamical behaviors and yields deeper insights into complex ecological systems.

Some of the most computationally demanding problems in science and engineering involve large-scale and high-dimensional matrix and tensor differential equations (MDEs and TDEs). These arise in applications ranging from the Schr¨odinger equation and kinetic transport mod- els (Fokker–Planck and Boltzmann equations) to probability density function transport in turbulent combustion and Hamilton–Jacobi–Bellman equations. Even in three-dimensional settings, such as turbulent reacting flows, the sheer scale and multiscale structure of the dynamics render direct numerical treatment prohibitively expensive. The underlying chal- lenge is structural: the number of degrees of freedom grows combinatorially with dimension, resolution, or parametric complexity. Low-rank matrix and their extension to tensor network approximations provide a principled framework to address this challenge by exploiting in- trinsic low-dimensional structure and multi-dimensional correlations in the solution. In this presentation, we focus on computational strategies for solving nonlinear MDEs and TDEs directly in adaptive low-rank form. We introduce a novel formulation based on adaptive cross interpolation and develop CUR-type decompositions grounded in the Discrete Em- pirical Interpolation Method (DEIM) for matrix, tensor-train, and Tucker representations. These DEIM-CUR algorithms enable on-the-fly reduced-order modeling with near-optimal accuracy for broad classes of nonlinear dynamical systems. We demonstrate the effectiveness of these approaches across applications in turbulent com- bustion, sensitivity analysis, uncertainty quantification, and both high-dimensional and large-scale partial differential equations.

Bio: Hessam Babaee is an Associate Professor in the Department of Mechanical Engineering and Materials Science at the University of Pittsburgh. He earned his Ph.D. in Mechanical Engineering and an M.S. in Applied Mathematics from Louisiana State University in 2013. After completing postdoctoral research in the Department of Mechanical Engineering at MIT, he joined the University of Pittsburgh in January 2017. His research develops low- rank approximation methods for high-dimensional dynamical systems, with an emphasis on reduced-order modeling using time-dependent bases and applications to flow stability and turbulent combustion. His work has been supported by the National Aeronautics and Space Administration (NASA), the National Science Foundation (NSF), the National Institutes of Health (NIH), the Department of Energy (DOE), the Air Force Office of Scientific Research (AFOSR), and national laboratories including Los Alamos and Sandia.

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