### Course Description:

In recent decades topological methods have been used in order to tackle many problems in combinatorics. One such example is Lovasz's proof of Kneser's conjecture from 1978. On the other hand, there are several situations in topology where combinatorial constructions hold the key to effective understanding. In this course we explore some material of these two kinds. Namely, our focus will be on (1) the study of the combinatorial properties or structure of topological objects and (2) using topological techniques to answer combinatorial problems. Topics such as topological graph theory, piecewise-linear (PL) topology, subspace arrangements,
and discrete applications of Borsuk-Ulam theorem will be treated.

### References and Additional Reading:

#### Main References:

J. Matousek, Using the Borsuk-Ulam Theorem, *Lectures on Topological Methods in Combinatorics and Geometry,* Springer, 2008.
D. Kozlov, Combinatorial Algebraic Topology, *Algorithms and Computation in Mathematics,* Springer, 2008.
C. Rourke and B. Sanderson, Introduction to Piecewise-Linear Topology, Springer, 1972.
#### Additional Reading:

A. Björner, Topological Methods, In: *Handbook of Combinatorics*.
M. de Longueville, A Course in Topological Combinatorics, Springer, 2013.
A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, , Oriented Matroids, Cambridge University Press, 2009.
P. Orlik, Introduction to Arrangements, CBMS Regional Conference Series in Mathematics, 1989.
M. Wachs, Poset Topology: Tools and Applications.
Jordan Curve Theorem (preliminary version).

### Lecturer:

Afshin Goodarzi

Department of Mathematics KTH, Stockholm

**Visiting address:** Lindstedtsvägen 25

Room number: 3437

**Email:** afshingo@kth.se

### Prerequisites:

We will assume a knowledge of elementary facts in algebra and topology. Other than that, some
familiarity with basic concepts from algebraic topology would be useful but not necessary.

### Examination:

The course has 7.5 credit.
The grading is on the basis of four sets of homework assignments, and a written report on a subject related to the course.

### Homework sets:

Homework #1.

Homework #2.