This lecture will take place online in RWTH Moodle and via Zoom.
Polyhedral Methods in Algebra
In the summer term, the lecture Polyhedral Methods in Algebra will be given by Christian Steinert for 5 credit points (3h per week). The lecture is designed for Bachelor and Master students of mathematics. The content of the lectures Linear Algebra I and II is required.
It is a fundamental principle of mathematics to translate problems from one area of research to another area and solve them there. It has proved especially fruitful to tackle algebraic problems with methods from polyhedral geometry. We want to explore this connection in selected examples – reaching stateoftheart research, hence preparing the participants of the lecture to read scientific articles and write thesis papers.
After an introduction to the theory of polytopes, where we will meet many known objects from school and everyday life, we will play the game magic squares to solve eigenvalue problems of hermitian matrices. Furthermore, we will follow Newton’s steps and compute intersection points of polynomials as volumes of polytopes. In our third application we will share some light on the tired quest of paying the morning bun at your local bakery with cash. This will lead us to a beautiful theory due to Eugène Ehrhart, whose applications in algebraic geometry and representation theory are currently being studied by members of the ART chair.
Time and place of the lecture will be announced at a later date. Registration will be done via RWTHonline at the beginning of the semester. Upon registration one automatically enters the learning space in RWTHmoodle, where exercises will be organized and further information can be found.
Literature

Lecture Script

Günter M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, SpringerVerlag, New York, 1995.

Branko Grünbaum, Convex Polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer, New York, 2003, editiert von Volker Kaibel, Victor Klee and Günter M. Ziegler.

Matthias Beck und Sinai Robins, Computing the Continuous Discretely, second ed., Undergraduate Texts in Mathematics, Springer, New York, 2015.