Towards Mordell’s Theorem: The Finiteness of E(Q)/2E(Q)

In this seminar paper we prove the finiteness of E(Q)/2E(Q). Here, E(Q) denotes the group of rational points of
an elliptic curve E over Q. These statements can be used to prove Mordell’s Theorem that E(Q) is finitely
generated. Structure and methods are according to Knapp: “Elliptic Curves”, Chapter IV, Sections 3, 4 and 9.

The seminar “Elliptic Curves” was held by Prof. Nils Scheithauer at TU Darmstadt in Winter Term 2010/2011.

Towards Mordell’s Theorem: The Finiteness of E(Q)/2E(Q)

Locally Compact Groups: Isomorphism Theorems & Cyclic Subgroups

This treatise was prepared for the seminar Locally Compact Groups held by PD. Dr. Ralf Gramlich and Stefan Witzel in August 2010 at TU Darmstadt. The seminar was structured according to Markus Stroppel’s book.

In Section 1 we prove several classical isomorphism theorems for topological groups. Furthermore, we
state sufficient criteria for a topological group to be isomorphic to an inner direct product. In order to do
so, we will need an open mapping theorem for topological groups which yields that every surjective morphism between
topological groups is open if the groups satisfy certain compactness properties.

We proceed in Section 2 by analyzing the structure of certain locally compact groups based on their
subgroups. Weil’s Lemma consists of two structure results for locally compact Hausdorff groups $G$. In
particular, for each $g \in G$ the cyclic group $\langle g \rangle$ is either discrete and infinite or has compact
closure in $G$. We continue by classifying certain Abelian topological groups as direct products of a free
Abelian group with an open subgroup. Additionally, we state an existence criterion for discrete subgroups of locally
compact Abelian Hausdorff groups. Finally, we give some results of purely algebraic nature.

Locally Compact Groups: Isomorphism Theorems & Cyclic Subgroup

As of March 9, 2011, there is some further material available:

• Seminar Book (seminar papers of all participants, partially in German)
• An Introduction to Topology (notes by S. Witzel)
• The Modular Function of a Haar Measure (notes by A. Mars, in German).

Nets in Metric Spaces

In topology, nets generalize the concept of sequences by allowing the index set to be an arbitrary so-called directed set. We give an elementary introduction of metric-space valued nets concerning their convergence theory and applications to real calculus. Also, we give short remarks on further generalizations when dealing with topological spaces. The treatise was prepared for a talk in the introductory seminar “Topological Spaces” held by Prof. Dr. Steffen Roch in December 2009 at TU Darmstadt. It is available only in German.

German abstract:
Netze sind Verallgemeinerungen von Folgen. Ihre Konvergenztheorie bietet gewisse Analogien zu der von Folgen, liefert aber im Kontext allgemeiner metrischer und insbesondere topologischer Räume grundlegendere strukturelle Einsichten. Im folgenden zeigen wir elementare Resultate über Netzkonvergenz in metrischen Räumen und einige Anwendungen in der reellen Analysis. An geeigneten Stellen geben wir einen Ausblick auf das Verhalten von Netzen in allgemeinen topologischen Räumen.
Dies ist eine Ausarbeitung zu dem Proseminar “Topological Spaces” bei Prof. Dr. Steffen Roch im Wintersemester 2009/2010 an der Technischen Universität Darmstadt.

Netze in metrischen Räumen

An Epistemological Reflection on Mathematics

A treatise on the philosophy of mathematics in Kant’s Transcendental Aesthetic in relation to the logicist critique by Frege and Russell. This has been the final paper for my Abitur in the subject of Ethics written in early 2008. It is only available in German.

German abstract:
Diese Jahresarbeit diskutiert epistemologische und ontologische Interpretationen der Mathematik. Insbesondere zählen dazu Kants transzendentale Ästhetik in der Elementarlehre, die sogenannte Grundlagenkrise der Mathematik zu Beginn des 20. Jahrhunderts und der Bezug der Mathematik zu den Natur- und Geisteswissenschaften. Abgedeckt werden dabei Aspekte der vom Lehrplan des Hessischen Kultusministeriums vorgeschriebenen Themen “Anthropologische Grundannahmen” (12.1), “Konkurrierende Normbegründungen in der [philosophischen] Tradition” (12.2), “Vernunft” (12.1/2), “Naturbegriff” (13.1) sowie das fakultative Thema “Wissenschaftstheorie” (12.1).

Eine erkenntnistheoretische Betrachtung der Mathematik