The course offers an introduction in solving problems in combinatorics and discrete geometry by topological methods with special emphasis on the Borsuk-Ulam theorem and its generalizations. Some main results to be covered are the solution of Kneser's conjecture, and the van Kampen-Flores theorem. The aim is to make some of the elementary topological methods more easily accessible to non-specialists in topology. Background in undergraduate mathematics is assumed, as well as a certain mathematical maturity, but no prior knowledge of algebraic topology.

The class meets Monday, Wednesday, and Friday from 14:00 to 14:50 in room 2443 in the Computer Science Building (E3-1). Lectures are given in English.

Students will be graded based on homeworks and a quiz.

We will use the following textbook:

- Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry (Universitext), Jiri Matousek.

Here is a rough list of what we will cover in each week of the semester.

Week 1 | Introduction |

Week 2 | Topological spaces |

Week 3 | Simplicial complexes |

Week 4 | Abstract simplicial complexes |

Week 5 | The Borsuk-Ulam theorem |

Week 6 | Ham-Sandwich theorem |

Week 7 | Kneser's Conjecture |

Week 8 | Midterm exam week |

Week 9 | Dolnikov's Theorem |

Week 10 | Gale's Lemma |

Week 11 | Nonembeddability theorems |

Week 12 | Deleted products and joins |

Week 13 | The Van Kampen-Flores Theorem |

Week 14 | G-Spaces |

Week 15 | The Topological Tverberg Theorem |

Week 16 | Final Exam week |

The material covered in the lectures so far:

02-11 | Introduction | |

02-13 | Simplicial complexes | |

02-15 | Simplicial mappings | |

02-18 | Chain mappings, odd and even maps | |

02-20 | Borsuk-Ulam in two dimensions | |

02-22 | Sperner's Lemma, general Borsuk-Ulam | |

02-25 | Brouwer's fixpoint theorem from Sperner's lemma | |

02-27 | Ham-Sandwich cut, Necklace | |

02-29 | No class | |

03-03 | Necklace theorem, Kneser's theorem | |

03-05 | Partitioning a measure with two lines, review | |

03-07 | Colored caratheorody | lecture notes |

03-10 | Quotient space | |

03-12 | Join | |

03-14 | Partitioning a measure by 3 coincident lines | |

03-17 | Z_{2}-space and Z_{2}-maps | |

03-19 | The Z_{2}-index | |

03-21 | No class | |

03-24 | The Z_{2}-index (cont.) | |

03-26 | Z_{2}-index and non-embeddability | |

03-28 | No class | |

Midterm week | ||

04-07 | Bier-spheres | de Longueville's proof |

04-09 | Holiday | |

04-11 | Sarkaria's inequality | |

04-14 | Sarkaria's coloring theorem | |

04-16 | Four points form a rectangle | |

04-18 | Homework discussion | |

04-21 | Box-complex, introduction to G-spaces | |

04-23 | ||

04-25 | ||

04-28 | No class | |

04-30 | No class | |

05-02 | No class | |

05-05 | Holiday | |

05-07 | ||

05-09 | ||

05-12 | Holiday | |

05-14 | Quiz | |

05-16 | ||

05-19 | ||

05-21 | ||

05-23 |

There will be a few homework assignments in this course. You will
have about *one to two weeks* to complete an assignment.
Homeworks must be submitted *on paper* and must be handed in at
the beginning of class.

- Homework due April 11.

We have a bulletin board for announcements and discussions. You can also post your questions there. Both Korean and English are acceptable on the BBS :-)

Otfried Cheong