Two Bachelor's Degree Project Presentations in Mathematics

Time: Harald Karlstedt
Student: 10:15 - 12:00
Title: A bridge to Plateau Problem

Abstract: Plateau's problem is a famous mathematical problem of finding a minimal surface (a "soap film") with a given boundary. In this thesis, we discuss a modern approach to this problem using techniques from Geometric Measure Theory. As a first step, we define the notion of a current, a linear functional on the space of smooth differential forms with compact support, which may be seen as a natural generalization of an oriented manifold with locally finite Hausdorff measure, since such manifolds can "act" on forms in a continuous way. We then focus on the class of rectifiable currents that has a useful compactness theorem not requiring uniform curvature bounds. This compactness theorem is then applied to prove the existence of solutions to a version of Plateau's problem.

Time: 11.15 - 12.00
Student: Tim Erixon
Title: A Study Into the Relationship Between Treewidth and Pathwidth

Abstract: In this paper we go on to explain the tree-and-pathwidth of a simple, connected, undirected graph depending on its decomposition. We then go on to exemplify their different use cases by showing how they can be applied to solve different famously hard graph theoretic problems, and comparisons are made to evaluate these parameters. We find in the end that both parameters have their uses, and that interestingly enough they both have different ways of characterising these parameters, outside their respective decompositions.