MAT773 Topics in convex geometry (HS22)

Lecturer: Dr. Johannes Schmitt
Time and Place: Friday, 10:15 - 12:00, Room Y27H25
Further information can be found in the UZH Course Catalogue.

Content

A polytope is the convex hull of finitely many points in real d- dimensional space. Examples include triangles in the plane, the five platonic solids in three-dimensional space or the so-called permutohedron in d dimensions. Polytopes and other convex objects like cones and polyhedra appear in many areas of mathematics. And even though their definition is simple, it is possible to ask many interesting questions about them, such as:

Can every graph be realized as the set of vertices and edges of a 3- dimensional polytope?
How does the number of integer points inside a polytope behave when we scale it with some positive integer t?
What is a fast algorithm for computing the convex hull of a given set of points?

In the seminar, we will start with introducing the basic notions and results of convex geometry. Based on this foundation, we then have a look at more specialized questions such as the ones described above (in particular concerning their combinatorics and computational approaches). The precise questions can depend on the interests of the participants.

List of talks

Here you find a preliminary list of topics for the talks.
Here is the preliminary assignment of topics and dates:

Topic	Date	Student
Basic definitions	23.09	Fadil Osmani
Further examples and main theorems	30.09	Argetim Beluli
Proofs of main theorems	07.10	Cagri Camoglu
Farkas lemmas and further basic results	14.10	Yonatan Christoph
Faces of polytopes	21.10	Jenny Pletscher
Polarity and the representation theorem for polytopes	28.10	Blerim Alimehaj
Platonic solids	04.11	Anna Glapka
Linear programming and related concepts	11.11	Weronika Wawrzyniak
More on graphs and polytopes	18.11	Philine Schönenberger
Polyhedral complexes and Schlegel diagrams	25.11	Chiara Romano
Review and exercise session	02.12	Chen Ping
The simplex algorithm	09.12	Özgür Özsu
Further algorithms and practical implementations	16.12	Delia Schüpbach
Ehrhart theory	23.12	Mattia Bottoni

Requirement for passing

In order to pass the seminar and get credit points, you have to:

give a talk from the above list to the remaining participants; the length of the talk should be about 90 minutes, the language of the talk is English
hand in a written part towards the end of the semester; this should be written in LaTeX and can be a summary of the talk or solutions to exercises in the part of the script you are discussing
attend (most of) the talks of the other participants

There is no grade for the seminar, so it is simply pass/fail.

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