Instructor: András Szenes

Text: Notes by László Fehér available here, or may be bought at the office.
         You may find the book of Munkres (Topology) useful.

Prerequisite: Calculus, especially metric spaces, the notion of continuity, basics of set theory, definition and basic properties of groups.

Course description:  We start with point set topology, an essential part of modern analysis. We continue with the classification of surfaces, and finish with the fundamentals of algebraic topology, which is one of the most dramatic topics in modern geometry. The level will depend on the background of the class.

Topics:

• Experiments in Topology: Untie knots, Set yourself free, Spinning plates of Dirac, The chess problem the computer couldn't solve...
• The notion of continuity
• Metric spaces and continuous functions
• Open and closed sets
• Equivalent metrics
• The notion of a topological space: Examples and constructions
• Translating the experiments into mathematics
• Separation axioms: the zoo
• Compactness
• Surfaces and other manifolds
• How to distinguish them? Euler characteristic, brushing the hedgehog, the coloring problem.
• The curve which fills the square. Why dimension is a topological notion?
• Connectivity
• Homotopy and the fundamental group
• Homology, higher homotopies