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:
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 | Z2-space and Z2-maps | |
| 03-19 | The Z2-index | |
| 03-21 | No class | |
| 03-24 | The Z2-index (cont.) | |
| 03-26 | Z2-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.
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 :-)