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.
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
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
Equipment
Literature
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
Grading scale
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
Opportunity to raise an approved grade via renewed examination
Examiner
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