Two Bachelor's Degree Project Presentations in Mathematics

Time: 13.15 - 14.00

Student: Jakob Hart

Title: The rectangular peg problem

Abstract: The rectangular peg problem asks whether, given any smooth Jordan curve and any rectangle, can we find a similar rectangle with vertices on the curve? The answer is yes. The proof uses a result from symplectic geometry, namely that there is no embedded Lagrangian Klein bottle in C^2. This thesis presentation gives a brief introduction to symplectic geometry, and then presents the most important points of the proof.

Time: 15.15 - 16.00

Student: Erik Schahine

Title: Statistical Error Bounds in the Deep Ritz Method for the p-Laplacian

Abstract: This thesis investigates the statistical error of the Deep Ritz Method (DRM) when applied to the p-Laplace partial differential equation (PDE), a nonlinear generalization of the classical Laplace equation. The DRM, a variational approach to solving PDEs leveraging deep neural networks, has shown promise in solving high dimensional and nonlinear PDEs, yet its theoretical error characteristics remain insufficiently understood. We focus on the statistical error introduced by the empirical approximation of the energy functional used during training. In this work, we derive a theoretical upper bound on this for the case $p\ge2$, quantifying its dependence on sample size, network capacity, and the properties of the underlying PDE. This analysis provides insight into the reliability and convergence behavior of deep learning-based variational solvers in nonlinear PDE settings.