PHDTE00012  Experimental Topology

Volume 2013/2014
Content

In this course, we will explore surfaces, higher-dimensional manifolds and more general complexes from a triangulation point of view.

Our main focus will be on

- constructions (of extremal or otherwise interesting examples),

- tools (e.g., discrete Morse theory, homology computation, finite group presentations),

- combinatorial properties (such as face numbers, colorings and valence conditions),

- algorithmic aspects (of how to compose/decompose and to recognize triangulations),

- enumerative methods (in an isomorphism-free way)

- and applications (in material sciences).

We will see that almost all algorithmic problems in topology are hard to solve or even are unsolvable -- yet we will discuss random approaches that allow to settle some of these problems.

Our constructions and examples will cover, among other, equivelar surfaces, Seifert 3-manifolds, knotted triangulations, homotopy 4-spheres, non-PL spheres, random complexes, valence restricted triangulations and polycrystalline materials such as metals and steel.

Learning Outcome
Knowledge: Theoretical background on the various topics of the course.
Skills: Usage of computer tools and experimental approaches to construct or analyze specific examples.
Competences: Ability to combine the theoritical background with applications.
Basic concepts from topology, combinatorics and algebra
5 lectures and 2 exercises per week
Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Oral examination without preparation time
Marking scale
passed/not passed
Censorship form
No external censorship
Two internal examiners
Criteria for exam assesment

The student must in a satisfactory way demonstrate that she/he has mastered the learning outcome of the course.

  • Category
  • Hours
  • Lectures
  • 35
  • Exercises
  • 14
  • Preparation
  • 157
  • Total
  • 206