Classification of Topological Surfaces

December 23, 2020

The following essay is part of my application to the TopMath Programm at TUM. We give a brief overview of the classification problem of manifolds in low dimension and introduce a further subdivision of the objects at hand. After that, we state the main theorem, which asserts that any closed, connected topological surface is homeomorphic to one of three explicit families, and consequently proceed with a proof which translates the topological data of a surface into combinatorial data, which lends itself much better to classification. Finally, we briefly go over some generalizations to higher dimensions and to less well have classes of surfaces.

I was kindly allowed to hold a talk based on this essay in the context of the M10 Oberseminar at TUM.

On the Yoneda Lemma in Category Theory

August 7, 2018

The following essay is part of my application to the TU Munich. Its aim is to briefly state and explain the Yoneda Lemma. The lemma is one of the (if not the single) most central ones in category theory as it provides a very useful correspondence between functors and hom-sets. Simultaneously, it is a powerful lemma in the construction of isomorphism theorems; most famously the Yoneda Embedding.

On the Fundamental Group of R/Q

December 16, 2020

In an algebraic-topological context, together with homology and cohomology, the fundamental group is among the first invariants of a topological space one should consider. In this exposition, we want to compute the fundamental group of R/Q and give a geometric interpretation of the space. We will firstly go over a strategy that seems promising but is ultimately misleading (which is essentially what makes the example interesting).