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Abstract: The classical calculus of pseudodifferential operators extends differential operators in a way that is suited to the construction of parametrices (pseudo-inverses) for elliptic operators. A fundamental consequence is that elliptic operators are Fredholm between appropriate Sobolev spaces on compact manifolds.

On manifolds with boundary, this implication no longer holds. Melrose’s calculus of b-pseudodifferential operators is the analogous class of operators which leads to Fredholm properties for elliptic operators satisfying a certain condition related to the boundary. In this talk, I will compare the classical case and the boundary case, emphasizing the new features introduced by the boundary—most notably the b-stretched product and the indicial operator—and explain how these lead to Fredholmness on weighted b-Sobolev spaces.

Abstract: In chemical kinetics, accurately modeling the dynamic behavior of chemical systems is essential for predicting reaction outcomes and optimizing processes. However, the challenge known as the “curse of dimensionality” has posed significant difficulties for conventional techniques employed in addressing the chemical master equation (CME). This predicament arises when the state space of the Markov chain expands exponentially with the number of species, rendering the CME computation practically unsolvable.

In this talk, I will discuss some methods of solving the CME, including Gillespie’s algorithm, the Chemical Langevin Equation, and the Method of Moments, along with an overview of tensor train and machine learning-based methods, which offer promising strategies for gaining insights into complex biological systems.

Abstract: Cell migration is a fundamental process in physiology and disease, yet it poses challenging problems in multiscale modeling and continuum mechanics. Cell motility arises from the coupling of intracellular transport, active force generation, and evolving geometry. Cytoskeletal dynamics, in particular actin turnover and force production, provides a rich setting for mathematical analysis. In this talk, I will present a mathematical framework for mammalian cell motility based on multiphase continuum models with moving boundaries. The formulation incorporates fluid-structure interaction and active stresses to describe the coupled evolution of cytoskeletal flow and cell shape. The model predicts how migration efficiency depends on actin dynamics and geometric features of the cell. If time permits, I will also present a mechanical-electrical-chemical coupled model for water-driven cell motility induced by polarized membrane ion transport. This second framework highlights how transport processes and force balance together generate directed motion.

Biography of the speaker: Yizeng Li is an Assistant Professor in the Department of Biomedical Engineering at Binghamton University. She received MS from Mathematics and PhD from the Department of Mechanical Engineering at the University of Michigan-Ann Arbor. Afterwards, she was a postdoctoral researcher at Johns Hopkins University's Department of Mechanical Engineering and Institute for NanoBioTechnology. Her backgrounds are in theoretical mechanics and applied mathematics with applications to biophysics and mechanobiology. Li develops physiology-based mathematical models for cell motility, polarization, volume regulation, electro-homeostasis, signal transduction, and other biophysics problems. She also combines mathematical models with experimental data to explain non-intuitive cell biology phenomena.

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Abstract: The nonlinear shallow water equations are used to model the free surface flow in rivers and coastal areas for which the horizontal length scale is much greater than the vertical length scale. They have wide applications in oceanic sciences and hydraulic engineering. In this talk, we study a numerical artifact of solving the shallow water equations over a discontinuous riverbed. For various first-order methods, we report that the numerical solution will form a spurious spike in the numerical momentum at the discontinuous point of the bottom. This artifact will cause the convergence to a wrong solution in many test cases. We present a convergence analysis to show that this numerical artifact is caused by the numerical viscosity imposed at the discontinuous point. Motivated by our analysis, we propose a numerical fix which works for the nontransonic problems.

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Abstract: Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) with broad applications in physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is to learn the associated Green’s function. However, Green’s functions are notoriously difficult to approximate due to their poor regularity, often requiring large neural networks and long training times.

In this talk, we address these challenges by leveraging multiscale neural networks to learn Green’s functions efficiently. Through theoretical analysis based on multiscale Barron space techniques, together with numerical experiments, we show that the multiscale approach significantly reduces the required network size and accelerates training. We then extend this framework to operator learning, enabling neural networks to efficiently and accurately learn the mapping from coefficient functions to Green’s functions.

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