In this talk, we introduce a generalized Schauder theory for degenerate and singular parabolic equations. The key idea is an approximation scheme based on fractional-order polynomials—s-polynomials—which replace constant coefficients in the classical setting. This approach not only recovers the classical regularity results for uniformly parabolic equations but also extends to operators where traditional bootstrap arguments are difficult to apply.

Neural networks have become increasingly effective for approximating solutions to partial differential equations (PDEs). This talk presents three advances that improve both accuracy and computational efficiency. First, I introduce an augmented Lagrangian formulation of the physics-informed loss that strengthens constraint enforcement and improves accuracy near domain boundaries. Second, I develop efficient architectures based on hypernetworks and graph neural networks that learn PDE solution operators with markedly small model sizes. Finally, I describe Neural-Galerkin schemes with low rank approximations for operator learning, which achieves a favorable accuracy-efficiency trade-off.

In this presentation, we discuss recent existence results for nonlinear diffusion equations with a divergence-type drift term, which are broadly applicable to various reaction-diffusion equations, including Keller-Segel models. We focus on identifying appropriate functional spaces for the drift, guided by the nonlinear diffusion and initial data. Using techniques from the theory of Wasserstein spaces, we construct weak solutions and establish their regularity properties.

We study the partial dimensional semi-classical Weyl’s laws, describing the quantum subband structures for two-dimensional electron gases (2DEGs). As a simple application, we derive lowest free energy states for the subband models describing non-interacting 2DEGs.