# FSF3705 Polytope Theory 7.5 credits

A convex polytope is the convex hull of finitely many points in a Euclidean space of finite dimension. They are important objects in combinatorics, algebra and optimisation. The course will mainly study polytopes from a combinatorial perspective.

Course offerings are missing for current or upcoming semesters.
Headings with content from the Course syllabus FSF3705 (Spring 2022–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

Basic facts about polytopes and methods to study them, e.g.

• Projections, face lattice, shelling, f-vector, triangulations, Ehrhart polynomial, lattice polytopes, h*-polynomial, secondary polytope

The course concerns also many beautiful and important constructions of special polytopes, e.g.:

cyclic polytope, Birkhoff polytope, zonotope, Minkowski sum, 0/1-polytope, transportation polytope, permutahedron, associahedron

### Intended learning outcomes

The course will give a basic knowledge of theory and methods in the theory of convex polytopes. The goal is to give a good and deep knowledge as a firm ground both for further studies in mathematics and for applications in other disciplines. More specifically the student should after the course

• know basic concepts and terminology in the theory for convex polytopes.
• know and be able to use important special polytopes and methods to construct new ones.
• be able to interpret combinatorial properties of a polytope from its face lattice
• know the basic problems and ideas in Ehrhart theory, and learn some techniques to compute related invariants of lattice polytopes.
• have increased intuition about properties of polytopes in higher dimensions and have realised how easy it is to guess incorrectly about properties in dimensions higher than 3.

### Course disposition

No information inserted

## Literature and preparations

### Specific prerequisites

A Master degree including at least 30 university credits (hp) in in Mathematics (including SF1631 Discrete mathematics and SF1624 Linear algebra or similar knowledge).

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

No information inserted

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• INL1 - Assignment, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The examination consists of: Hand-in problems, a written exam, and a presentation of a scientific text

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web FSF3705

### Main field of study

This course does not belong to any Main field of study.

Third cycle