Algebraic Topology

Fact: If you have a dog which is completely covered in hair, then there is no way of combing that hair smoothly, so that there is no parting or bald spot. This is the so-called `hairy dog theorem'.
Fact: no matter how badly you make a sandwich out of two pieces of bread and a slice of ham, it is always possible to find a plane cutting the sandwich which bisects exactly each piece of bread and the slice of meat. This is the so-called `ham-sandwich theorem'.
Fact: if you associate to each point of the Earth's surface the two numbers t and p given by temperature and air pressure at this point, then there is always at least one point which has the same values for t and p as its diametrically opposite one. This is the so-called `Borsuk-Ulam theorem'. These are all very deep geometric facts about the shapes of dogs, ham sandwiches and the Earth; they are especially deep as they are true for any shape or size of dog,
sandwich or planet, and they are all proved by a remarkably
successful set of mathematical ideas and methods that date from the early mid of last century. These ideas and ones like them constitute the subject Algebraic Topology. The basic idea of the subject is to find a formal way to translate geometric problems into an appropriate algebraic language. If this is done successfully, then the geometric problem is usually reduced to a fairly simple piece of algebra and can be solved by algebraic means. In this course we provide the technical tools (homology groups and cohomology groups of
topological spaces) and will discuss several geometric applications.
In particular we shall prove each of the facts above by
translating the underlying geometry into questions about integers, groups or vector spaces.

The (co)homological methods of algebraic topology have later been used in many other parts of mathematics, for example in algebraic geometry and group theory, and are now a standard tool in these subjects.Thus, the course will be of interest to a wide variety of master students.