Bachelor- und Masterarbeiten Wintersemester 2024/2025

The Truncated Birkhoff Completion

In the Birkhoff completion, certain concepts, especially in the lower part of the lattice, are generated by new objects derived from implications. However, these objects may introduce inconsistent or unrealistic combinations of attributes. Truncating the lattice by eliminating such inconsistencies ensures a more meaningful and applicable structure, particularly in real-world scenarios. This project aims to study the truncated Birkhoff completion as a method for addressing inconsistencies that arise in lattice structures generated through the Birkhoff completion of non-distributive lattices.

Inquiries: Mohammad Abdulla

Formalizing Results of Formal Concept Analysis

Formalization of a large number of definitions and theorems in Algebra, Number Theory and Analysis in the lean prover brings the field of formal theorem proving to the forefront of mathematical research. However, mathematical notions and results from the field of Formal Concept Analysis are not yet included. Thus the goal of this project (or thesis) is to formally prove the “Basic Theorem on Concept Lattices” (or a comparable result) in order to provide a stepping stone in this direction. Essentially, this involves building core definitions and necessary preliminary results by extending existing ones. We will aim to build a lean blueprint and if you are only interested in a small project, we will find a reasonable partial realization to stop at.

Inquiries: Tobias Hille

Formal Concept Analysis mit Attribut und Objektordnungen

In dieser Arbeit untersuchen Sie, inwiefern sich die Theorie der formalen Begriffsanalyse auf den Fall übertragen lässt, dass wir eine lineare Ordnung auf den Attributen und den Objekten vorliegen haben.

Das Ziel ist es, die in der FCA üblichen Ideen (Begriffe, Implikationen etc.) auf solche Datensätze zu übertragen und die Theorie mit Echtwelt-Datensätzen zu evaluieren.

Informationen: Dominik Dürrschnabel

Invariants of Formal Contexts

It is not easy to recognise whether two (reduced) formal contexts are isomorphic, or given a set of formal contexts, how many different formal contexts are contained there. One aid are invariants, i.e. derived quantities, that do not depend on the concrete representation of the formal context. Simple examples are the number of attributes of the context or the number of objects of the context. If two contexts have different values for an invariant, the contexts are not isomorphic. The aim is to examine formal contexts with regard to possible invariants. Formal contexts can be represented as bipartite graphs, therefore, known graph invariants in particular are to be considered.

Inquiries: Tobias Hille

Detecting Graphs in Images

The project aims to develop a machine learning model that can detect (simple) graphs in images. This involves not only an extensive literature review but also gathering useful training data. Moreover we need to train the model to recognize and segment images containing graphs. The project will use image classification algorithms and techniques to achieve this goal. Completing individual parts may already be enough for a successful conclusion. You will build upon work done by previous participants.
Most (if not all) of the programming will be done in Python.

Inquiries: Tobias Hille

Generators for and Properties of random bipartite Graphs

In this project, we will conduct a practical investigation into the random generation of bipartite graphs. We will build upon previous works in the field and analyze the properties of the produced distributions. Additionally, we will simulate real-world data, potentially using approaches like GAN training.
Most (if not all) of the programming will be done in Python.

Inquiries: Tobias Hille

Stabilität von Formalen Kontexten

Wir nennen einen formalen Kontext stabil, wenn sich beim Setzen oder Entfernen jedes Kreuzes die Größe des Begriffsverbandes nicht verkleinert. Untersuchen Sie Echtweltdaten auf Stabilität und untersuchen Sie, inwiefern sich die Stabilität als Bewertungsmaß für intrinsisch sinnvolle Daten eignet.

Informationen: Dominik Dürrschnabel

Intrinsic Triangulation of Loss Landscapes of Neural Networks

Recent work investigating geometry and topology of loss landscapes of neural networks revealed interesting properties regarding connectivity of clusters of local minima [1], [2]. In this work you will try to apply advances in computation intrinsic triangulation for 3d rendering [3] towards those surfaces (or reasonable hyperplane projections). One possible motivation for this is the change for an ability to construct geodesic paths on the constructed approximation.
Most (if not all) of the programming will be done in Python.

Inquiries: Tobias Hille