Three Bachelor's Degree Project Presentations in Mathematics

Time: 10:15 - 12:00

Speaker: Arvid Stenarson

Title: An Introduction to Riemann Surfaces: From Multivalued Functions to Riemann-Roch

Abstract: Riemann Surfaces is the innate domain of meromorphic functions. Historically, they arose when dealing with analytic continuation and multi-valued functions in the complex plane and is today of great interest in the field of string theory. This thesis begins by heuristically studying the construction of elementary Riemann surfaces in the historical context they were first thought of. It then moves on to introduce the basics of the more general theory, talking about abstract Riemann surfaces, differential forms and sheaf cohomologies. A combination of complex analysis, topology and algebraic tools culminates in the proof of the Riemann-Roch theorem connecting local analytic results with global topological properties.

Time: 11.15 - 12.00

Speaker: Hugo Eklöf

Title: Can one hear the area of a drum?

Abstract: This thesis focuses on proving Weyl's law for arbitrary, two-dimensional domains. The title is inspired by the expression popularized by Mark Kac in his article "Can one hear the shape of a drum?" The considered domains are characterized such that they resemble the physical structure of membranes with clamped endpoints, where the boundary can attain any shape as long as it is smooth. By considering the Wave equation for the perpendicular displacements of any point of a domain, it is derived that the constituted homogeneous Dirichlet eigenvalue problem yields a spectra of eigenvalues that is correlated to the domain's vibrational frequencies. Consequently, some of the classical results of the Laplacian and the related eigenfunctions and eigenvalues are highlighted and then utilized in the context of variational calculus to find upper and lower bounds to spectra of different shapes. The homogeneous Dirichlet eigenvalue problem is also solved for some particular shapes known to have explicit solutions. These results are then used to approximate arbitrary domains and to lastly prove Weyl's law, which describes an asymptotic relationship between the spectra and the area of a domain. While the result can be extended to other dimensions, this thesis only answers the question in the affirmative for the two-dimensional case.

Time: 13.15 - 14.00

Speaker: Jakob Lynas

Title: Connections on Vector Bundles and the Berry Phase

Abstract: The Berry phase is a physical phenomena which has seen considerable relevance in physics since its modern formulation in 1984, and is also an interesting example of mathematical gauge theory being utilised in non-relativistic quantum mechanics. This thesis begins by introducing the notion of a vector bundle, as well as some basic geometric and topological constructions on them. After a brief discussion of basic quantum mechanics, the Berry phase is derived, and the calculation is undertaken for the case of a spin-1/2 particle in a homogeneous magnetic field.