Applied Math Seminar

Synopsis:

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools.  We focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction.  We introduce a minimal two-dimensional free-boundary PDE model that captures the evolution of the cell shape and nonlinear diffusion of myosin.

We first consider a linear diffusion model with two sources of nonlinearity: Keller-Segel cross-diffusion term and the free boundary that models moving/deformable cell membranes. We establish asymptotic linear stability and derive an explicit formula for the stability-determining eigenvalue. Next, we consider the effect of nonlinear myosin diffusion, which results in the change of the bifurcation type from super- to subcritical, and obtain an asymptotic representation of the bifurcation curve. This allows us to derive an explicit formula for the curvature at the bifurcation point that controls the bifurcation type. In the most recent work in progress with the Heidelberg biophysics group, we study the relation between various types of nonlinear diffusion and bistability. Finally, we discuss novel mathematical features of this free boundary  model with a focus on non-self-adjointness, which plays a key role in the stability analysis.  We conclude by presenting an example when  the presence of a spectral gap does not guarantee stability and contrast this example with non linear stability in our minimal model.

Joint works with A. Safsten & V. Rybalko (Transactions of AMS 2023, and Phys. Rev. E 2022),  with O. Krupchytskyi &T. Laux (Journal of nonlinear Scinence 2026), and with A. Safsten & L. Truskinovsky  (ARMA 2026), with A. Safsten (2026-in progress). This work has been supported by NSF grants DMS-2404546,  DMS-2005262, and  DMS-2404546.

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Adaptive gradient methods such as Adam and Adagrad are widely used in machine learning, yet their effect on the generalization of learned models – relative to methods like gradient descent – remains poorly understood. Prior work on binary classification suggests that Adam exhibits a “richness bias,” which can help it learn nonlinear decision boundaries closer to the Bayes-optimal decision boundary relative to gradient descent. However, the coordinate-wise preconditioning scheme employed by Adam renders the overall method sensitive to orthogonal transformations of feature space. We show that this sensitivity can manifest as a reversal of Adam’s competitive advantage: even small rotations of the underlying data distribution can make Adam forfeit its richness bias and converge to a linear decision boundary that is farther from the Bayes-optimal decision boundary than the one learned by gradient descent. To alleviate this issue, we show that a recently proposed reparameterization method – which applies an orthogonal transformation to the optimization objective – endows any first-order method with equivariance to data rotations, and we empirically demonstrate its ability to restore Adam’s bias towards rich decision boundaries. This is joint work with Adela DePavia and Vasileios Charisopoulos.

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