Wir wollen wir uns mit den Grundlagen der Mathematik
beschäftigen, die in den grundlegenden Veranstaltungen angesprochen wurden,
für deren umfassende Behandlung aber keine Zeit blieb. Wir werden dafür
wichtige Konzepte und bedeutende Sätze in den drei Bereichen Mengenlehre,
Logik und Kategorientheorie studieren. Wir werden uns mit den Axiomen der
modernen Mengenlehre beschäftigen, die Bedeutung (und die Mängel) von
Cantors Definition einer Menge untersuchen und das berühmte Auswahlaxiom
studieren. Wir werden verschiedene Zahlenräume konstruieren und Ordinal-
und Kardinalzahlen kennenlernen sowie verschiedene Stufen der Unendlichkeit
betrachten. Wir werden formale Sprachen untersuchen und uns mit der Logik
von mathematischen Beweisen beschäftigen und lernen, wie man diese formal
rechtfertigen kann.
Letztlich beschäftigen wir uns mit Kategorien. Viele Konzepte der Mathema-
tik trifft man in ähnlicher Form in verschiedenen Bereichen der Mathematik
wieder, denken Sie etwa an lineare Abbildungen, Gruppenhomomorphismen
oder stetige Abbildungen. Allen drei ist gemein, dass es sich um strukturer-
haltende Abbildungen handelt, obwohl die jeweiligen Strukturen kaum etwas
miteinander zu tun haben. Wir werden lernen, wie man solche Konzepte
unter einem Begriff subsumieren kann. Außerdem werden wir uns universelle
Konstruktionen anschauen und untersuchen, wie man Begriffe, die wir von
Mengen her kennen, in andere Gebiete der Mathematik sinnvoll übertragen
kann.

Voraussetzungen:
Keine Vorkenntnisse erforderlich, nur ein Interesse an logischen und formellen
Konzepten.
Die Vorlesung richtet sich an alle Studierenden, die ihr Verständnis der
Mathematik vertiefen und ein tieferes Verständnis der Grundlagen erlangen
möchten.

Are you fascinated by geometric shapes and their properties?
Are you an Bachelor's or Masters's student in mathematics looking to deepen your understanding of polytopes?
Then this seminar is for you!

In this seminar, we will delve into the fascinating world of polytopes, which are geometric objects that generalize the idea of a convex polygon or a convex polyhedron.
Polytopes have a wide range of applications in various fields, such as computer graphics, physics, and optimization.

We will cover a variety of topics related to polytopes, including:

• Basics of polytopes: We will begin by discussing the basic properties of polytopes. We will also explore the different types of polytopes, including simplices, cubes, and cross-polytopes.
• Combinatorial properties: We will delve into the combinatorial properties of polytopes, including the Euler characteristic, face numbers, and the Dehn-Sommerville equations.
• Algebraic properties: We will examine the algebraic properties of polytopes, including their symmetry groups and the representation of polytopes by linear inequalities.

- Applications: We will explore the different applications of polytopes in other areas of mathematics and science, such as linear programming, coding theory, and computer graphics.

• Computational methods: We will learn about computational methods for dealing with polytopes, including convex hull algorithms, triangulation algorithms and algorithms for counting the number of faces of a polytope.

Throughout the seminar, we will use examples and case studies to illustrate the concepts and theories discussed.
The seminar will also touch on the recent developments in the study of polytopes, including new results and open problems.

But this seminar is not just about listening and taking notes, we want you to actively participate in the class.
Each student is expected to give a talk on a specific topic related to polytopes, allowing you to not only deepen your own understanding but also share your knowledge with your peers.
This seminar is also a great opportunity for students to connect with other students with similar interests and to learn from each other.

Join us on a journey of discovery and exploration as we uncover the hidden beauty and complexity of polytopes.

Sign up now and be ready to be amazed!

A large number of problems arising in practical scenarios like communication, transportation, planning, logistics etc. can be formulated as discrete linear optimization problems. This course briefly introduces the theory of such problems. We develop a toolkit to model real-world problems as (discrete) linear programs. We also explore several ways to find integer solutions such as cutting planes, branch & bound, and column generation.

Throughout the course, we learn these skills by modeling and solving, for example, scheduling, packing, matching, routing, and network-design problems. We focus on translating practical examples into mixed-integer linear programs. We learn how to use solvers (such as CPLEX and Gurobi) and tailor the solution process to certain properties of the problem.

This course consists of two phases:

• One week Mon-Fri (full day) of lectures and practical labs in the end of September.
• An individual project period: One project has to be modeled, implemented, and solved individually or in a group of at most two students. The topic will be either developed with or provided by the lecturers. The project including the implementation has to be presented before the beginning of the winter semester.

There are no prerequisites except some basic programming skills to participate.